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We prove that if G is the circle group or a profinite group, then the all of the homotopical information of the category of rational G-spectra is captured by triangulated structure of the rational G-equivariant stable homotopy category. That is, for G profinite or S1, the rational G-equivariant stable homotopy category is rigid. For the case of profinite groups this rigidity comes from an intrinsic formality statement, so we carefully relate the notion of intrinsic formality of a differential graded algebra to rigidity. | Rational Equivariant Rigidity | 13,200 |
This paper develops a basic theory of H-groups. We introduce a special quotient of H-groups and extend some algebraic constructions of topological groups to the category of H-groups and H-maps. We use these constructions to prove some advantages in topological homotopy groups. Also, we present a family of spaces that their topological fundamental groups are indiscrete topological group and find out a family of spaces whose topological fundamental group is a topological group. | On H-groups and their applications to topological homotopy groups | 13,201 |
We show that if the fundamental groups of the complements of two line arrangements in the complex projective plane are isomorphic to the same direct sum of free groups, then the complements of the arrangements are homotopy equivalent. For any such arrangement, we construct another arrangement that is complexified-real, the intersection lattices of the arrangements are isomorphic, and the complements of the arrangements are diffeomorphic. | Line arrangements and direct sums of free groups | 13,202 |
The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical omega-groupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism. | Covering morphisms of crossed complexes and of cubical omega-groupoids
with connection are closed under tensor product | 13,203 |
We study the construction of tensor products of representations up to homotopy, which are the A-infinity version of ordinary representations. We provide formulas for the construction of tensor products of representations up to homotopy and of morphisms between them, and show that these formulas give the homotopy category a monoidal structure which is uniquely defined up to equivalence. | Tensor products of representations up to homotopy | 13,204 |
In the present paper we study a bordism theory related to pairs $(M,\, \xi),$ where $M$ is a closed smooth oriented manifold with a stably trivial normal bundle and $\xi$ is a virtual $\SU$-bundle of virtual dimension 1 over $M$. The main result is the calculation of the corresponding ring modulo torsion and the explicit description of its generators. | A bordism theory related to matrix Grassmannians | 13,205 |
For a 1-connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalences of $X$. The aim of this paper is to prove that, for every $n\in\Bbb N$, there exists a 1-connected rational CW-complex $X_{n}$ such that $\mathcal{E}(X_{n})\cong \underset{2^{n+1}\mathrm{. times}}{\underbrace{\Bbb Z_{2}\oplus... \Bbb \oplus \Bbb Z_{2}}}$. | Realizability of the group of rational self-homotopy equivalences | 13,206 |
In this note we define fibrations of topological stacks and establish their main properties. We prove various standard results about fibrations (fiber homotopy exact sequence, Leray-Serre and Eilenberg-Moore spectral sequences, etc.). We prove various criteria for a morphism of topological stacks to be a fibration, and use these to produce examples of fibrations. We prove that every morphism of topological stacks factors through a fibration and construct the homotopy fiber of a morphism of topological stacks. When restricted to topological spaces our notion of fibration coincides with the classical one. | Fibrations of topological stacks | 13,207 |
Let $X$ be a simply connected CW complex with finite rational cohomology. For the finite quotient set of rationalized orbit spaces of $X$ obtained by almost free toral actions, ${\mathcal T}_0(X)=\{[Y_i] \}$, induced by an equivalence relation based on rational toral ranks, we order as $[Y_i]<[Y_j]$ if there is a rationalized Borel fibration $Y_i\to Y_j\to BT^n_{\Q}$ for some $n>0$. It presents a variation of almost free toral actions on $X$. We consider about the Hasse diagram ${\mathcal H}(X)$ of the poset ${\mathcal T}_0(X)$, which makes a based graph $G{\mathcal H}(X)$, with some examples. Finally we will try to regard $G{\mathcal H}(X)$ as the 1-skeleton of a finite CW complex ${\mathcal T}(X)$ with base point $X_{\Q}$. | A Hasse diagram for rational toral ranks | 13,208 |
We study the continuous (co-)homology of towers of spectra, with emphasis on a tower with homotopy inverse limit the Tate construction X^{tG} on a G-spectrum X. When G=C_p is cyclic of prime order and X=B^p is the p-th smash power of a bounded below spectrum B with H_*(B) of finite type, we prove that (B^p)^{tC_p} is a topological model for the Singer construction R_+(H^*(B)) on H^*(B). There is a map epsilon_B : B --> (B^p)^{tC_p} inducing the Ext_A-equivalence epsilon : R_+(H^*(B)) --> H^*(B). Hence epsilon_B and the canonical map Gamma : (B^p)^{C_p} --> (B^p)^{hC_p} are p-adic equivalences. | The topological Singer construction | 13,209 |
We study the C_p-equivariant Tate construction on the topological Hochschild homology THH(B) of a symmetric ring spectrum B by relating it to a topological version R_+(B) of the Singer construction, extended by a natural circle action. This enables us to prove that the fixed and homotopy fixed point spectra of THH(B) are p-adically equivalent for B = MU and BP. This generalizes the classical C_p-equivariant Segal conjecture, which corresponds to the case B = S. | The Segal conjecture for topological Hochschild homology of complex
cobordism | 13,210 |
The goal of this paper is to set up an obstruction theory in the context of algebras over an operad and in the framework of differential graded modules over a field. Precisely, the problem we consider is the following: Suppose given two algebras A and B over an operad P and an algebra morphism from the homology of A to the homology of B. Can we realize this morphism as a morphism of P-algebras from A to B in the homotopy category? Also, if the realization exists, is it unique in the homotopy category? We identify obstruction cocycles for this problem, and notice that they live in the first two groups of operadic Gamma-cohomology. | Obstruction theory for algebras over an operad | 13,211 |
We prove two homotopy decomposition theorems for the loops on co-H-spaces, including a generalization of the Hilton-Milnor Theorem. These are applied to problems arising in algebra, representation theory, toric topology, and the study of quasi-symmetric functions. | Decompositions of looped co-H-spaces and applications | 13,212 |
In this paper we devote to spaces that are not homotopically hausdorff and study their covering spaces. We introduce the notion of small covering and prove that every small covering of $X$ is the universal covering in categorical sense. Also, we introduce the notion of semi-locally small loop space which is the necessary and sufficient condition for existence of universal cover for non-homotopically hausdorff spaces, equivalently existence of small covering spaces. Also, we prove that for semi-locally small loop spaces, $X$ is a small loop space if and only if every cover of $X$ is trivial if and only if $\pi_1^{top}(X)$ is an indiscrete topological group. | Small loop spaces and covering theory of non-homotopically Hausdorff
spaces | 13,213 |
We prove a conjecture of Kontsevich which states that if $A$ is an $E_{d-1}$ algebra then the Hochschild cohomology object of $A$ is the universal $E_d$ algebra acting on $A$. The notion of an $E_d$ algebra acting on an $E_{d-1}$ algebra was defined by Kontsevich using the swiss cheese operad of Voronov. The degree 0 and 1 pieces of the swiss cheese operad can be used to build a cofibrant model for $A$ as an $E_{d-1}-A$ module. The theorem amounts to the fact that the swiss cheese operad is generated up to homotopy by its degree 0 and 1 pieces. | Kontsevich's Swiss Cheese Conjecture | 13,214 |
In this paper we study the relationship between the moment-angle complex Z_k and the Davis-Januskiewicz space DJ(K) for a class of complexes K named missing-face complexes. If K has n vertices we consider the homotopy fibration sequence Z_k --> DJ(K) --> M where M is a product of n copies of infinite complex projective space. We observe that for such K, Z_k is homotopy equivalent to a wedge of spheres, and then show that under this equivalence the map Z_k --> DJ(K) is homotopic to a wedge sum of higher Whitehead products and iterated Whitehead products. | Higher Whitehead products in toric topology | 13,215 |
We introduce a new model for the secondary Steenrod algebra at the prime 2 which is both smaller and more accessible than the original construction of H.-J. Baues. We also explain how BP can be used to define a variant of the secondary Steenrod algebra at odd primes. | On the secondary Steenrod algebra | 13,216 |
In this paper we explore new relations between Algebraic Topology and the theory of Hopf Algebras. For an arbitrary topological space $X$, the loop space homology $H_*(\Omega\Sigma X; \coefZ)$ is a Hopf algebra. We introduce a new homotopy invariant of a topological space $X$ taking for its value the isomorphism class (over the integers) of the Hopf algebra $H_*(\Omega\Sigma X; \coefZ)$. This invariant is trivial if and only if the Hopf algebra $H_*(\Omega\Sigma X; \coefZ)$ is isomorphic to a Lie-Hopf algebra, that is, to a primitively generated Hopf algebra. We show that for a given $X$ these invariants are obstructions to the existence of a homotopy equivalence $\Sigma X\simeq \Sigma^2Y$ for some space $Y$. Further on, using the notion of Hopf algebras, we establish new structural properties of the cohomology ring, in particular, of the cup product. For example, using the fact that the suspension of a polyhedral product $X$ is a double suspension, we obtain a strong condition on the cohomology ring structure of $X$. This gives an important application in toric topology. For an algebra to be realised as the cohomology ring of a moment-angle manifold $\Z_P$ associated to a simple polytope $P$, we found an obstruction in the Hopf algebra $H_*(\Omega\Sigma \Z_P)$. In addition, we use homotopy decompositions to study particular Hopf algebras. | Hopf algebras and homotopy invariants | 13,217 |
We consider a topological integral transform of Bessel (concentric isospectral sets) type and Fourier (hyperplane isospectral sets) type, using the Euler characteristic as a measure. These transforms convert constructible $\zed$-valued functions to continuous $\real$-valued functions over a vector space. Core contributions include: the definition of the topological Bessel transform; a relationship in terms of the logarithmic blowup of the topological Fourier transform; and a novel Morse index formula for the transforms. We then apply the theory to problems of target reconstruction from enumerative sensor data, including localization and shape discrimination. This last application utilizes an extension of spatially variant apodization (SVA) to mitigate sidelobe phenomena. | Euler-Bessel and Euler-Fourier Transforms | 13,218 |
Normal maps between discrete groups $N\rightarrow G$ were characterized [FS] as those which induce a compatible topological group structure on the homotopy quotient $EN\times_N G$. Here we deal with topological group (or loop) maps $N\rightarrow G$ being normal in the same sense as above and hence forming a homotopical analogue to the inclusion of a topological normal subgroup in a reasonable way. We characterize these maps by a compatible simplicial loop space structure on $Bar_\bullet(N,G)$, invariant under homotopy monoidal functors, e.g. Localizations and Completions. In the course of characterizing homotopy normality, we define a notion of a "homotopy action" similar to an $A_{\infty}$ action on a space, but phrased in terms of Segal's 'special $\Delta-$spaces' and seem to be of importance on its own right. As an application of the invariance of normal maps, we give a very short proof to a theorem of Dwyer and Farjoun namely that a localization by a suspended map of a principal fibration of connected spaces is again principal. | Homotopy Normal Maps | 13,219 |
In this paper, we discuss the theory of quasi-fibrations in proper Bousfield localizations of model categories of simplicial sheaves. We provide a construction of fibrewise localization and use this construction to generalize a criterion for locality of fibre sequences due to Berrick and Dror Farjoun. The result allows a better understanding of unstable A^1-homotopy theory. | Fibre sequences and localization of simplicial sheaves | 13,220 |
By an $n$-Hawaiian like space $X$ we mean the natural inverse limit, $\displaystyle{\varprojlim (Y_i^{(n)},y_i^*)}$, where $(Y_i^{(n)},y_i^*)=\bigvee_{j\leq i}(X_j^{(n)},x_j^*)$ is the wedge of $X_j^{(n)}$'s in which $X_j^{(n)}$'s are $(n-1)$-connected, locally $(n-1)$-connected, $n$-semilocally simply connected and compact CW spaces. In this paper, first we show that the natural homomorphism $\displaystyle{\beta_n:\pi_n(X,*)\rightarrow \varprojlim \pi_n(Y_i^{(n)},y_i^*)}$ is bijection. Second, using this fact we prove that the topological $n$-homotopy group of an $n$-Hawaiian like space, $\pi_n^{top}(X,x^*)$, is a topological group for all $n\geq 2$ which is a partial answer to the open question whether $\pi_n^{top}(X,x^*)$ is a topological group for any space $X$ and $n\geq 1$. Moreover, we show that $\pi_n^{top}(X,x^*)$ is metrizable. | On Topological Homotopy Groups of $n$-Hawaiian like spaces | 13,221 |
A solution to the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on $\Z/2^{[3]}$--framed immersions, and quaternionic-cyclic structure on $\Z/2^{[4]}$--framed immersions. Using these concepts, we prove that for sufficiently large $n$, $n=2^{\ell}-2$, an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant. Additionally, for $\ell \ge 12$ (i.e., for $n \ge 4094$) an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant if this skew-framed immersion admits a compression of order 16. | Geometric approach to stable homotopy groups of spheres II. The Kervaire
invariant | 13,222 |
We find a simple algebraic model for rational G-equivariant spectra, where G is the p-adic integers, via a series of Quillen equivalences. This model, along with an Adams short exact sequence, will allow us to easily perform constructions and calculations. | Rational Z_p-Equivariant Spectra | 13,223 |
Given a finite quandle, we introduce a quandle homotopy invariant of knotted surfaces in the 4-sphere, modifying that of classical links. This invariant is valued in the third homotopy group of the quandle space, and is universal among the (generalized) quandle cocycle invariants. We compute the second and third homotopy groups, with respect to "regular Alexander quandles". As a corollary, any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of the Mochizuki 3-cocycle invariant. As another result, we determine the third quandle homology group of the dihedral quandle of odd order. | Quandle homotopy invariants of knotted surfaces | 13,224 |
We introduce the notion of a strongly homotopy-comultiplicative resolution of a module coalgebra over a chain Hopf algebra, which we apply to proving a comultiplicative enrichment of a well-known theorem of Moore concerning the homology of quotient spaces of group actions. The importance of our enriched version of Moore's theorem lies in its application to the construction of useful cochain algebra models for computing multiplicative structure in equivariant cohomology. In the special cases of homotopy orbits of circle actions on spaces and of group actions on simplicial sets, we obtain small, explicit cochain algebra models that we describe in detail. | Multiplicative structure in equivariant cohomology | 13,225 |
By expressing the geometric realization of simplicial sets and cyclic sets as filtered colimits, Drinfeld (arXiv:math/0304064v3) proved in a substantially simplified way the fundamental facts that geometric realization preserves finite limits, and that the group of orientation-preserving homeomorphisms of the interval [0,1] (resp. the circle R/Z) acts on the realization of a simplicial (resp. cyclic) set. In this paper, we first review Drinfeld's method and then introduce an analogous expression for the geometric realization of dihedral sets. We also see how these expressions lead to a clarified description of subdivisions of simplicial, cyclic, and dihedral sets. | On the geometric realization and subdivisions of dihedral sets | 13,226 |
Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than classical homotopy invariants on underlying spaces because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with temporal structure encoding the orientations of simplices and 1-cubes. In an attempt to develop calculational tools for directed homotopy theory, we prove appropriate simplicial and cubical approximation theorems. We consequently show that geometric realization induces an equivalence between weak homotopy diagram categories of cubical sets and directed spaces and that its right adjoint satisfies an excision theorem. Along the way, we give criteria for two different homotopy relations on directed maps in the literature to coincide. | Cubical approximation for directed topology | 13,227 |
We analyze the homological behavior of the attaching maps in the 2-local Goodwillie tower of the identity evaluated at S^1. We show that they exhibit the same homological behavior as the James-Hopf maps used by N. Kuhn to prove the 2-primary Whitehead conjecture. We use this to prove a calculus form of the Whitehead conjecture: the Whitehead sequence is a contracting homotopy for the Goodwillie tower of S^1 at the prime 2. | The Goodwillie tower for S^1 and Kuhn's theorem | 13,228 |
We extend the nonabelian Dold-Kan decomposition for simplicial groups of Carrasco and Cegarra in two ways. First, we show that the total order of the subgroups in their decomposition belongs to a family of total orders all giving rise to Dold-Kan decompositions. We exhibit a particular partial order such that the family is characterized as consisting of all total orders extending the partial order. Second, we consider symmetric-simplicial groups and show that, by using a specially chosen presentation of the category of symmetric-simplicial operators, new Dold-Kan decompositions exist which are algebraically much simpler than those of Carrasco and Cegarra in the sense that the commutator of two component subgroups lies in a single component subgroup. | Nonabelian Dold-Kan Decompositions for Simplicial and
Symmetric-Simplicial Groups | 13,229 |
The category Fin of symmetric-simplicial operators is obtained by enlarging the category Ord of monotonic functions between the sets {0,1,...,n} to include all functions between the same sets. Marco Grandis has given a presentation of Fin using the standard generators Ord (cofaces and codegeneracies) as well as the adjacent transpositions which generate the permutations in Fin. The purpose of this note is to establish an alternative presentation of Fin in which the codegeneracies are replaced by special maps which we call quasi-codegeneracies. We also prove a unique factorization theorem for products of cofaces and quasi-codegeneracies analogous to the standard unique factorizations in Ord. This presentation has been used by the author to construct symmetric hypercrossed complexes (to be published elsewhere) which are algebraic models for homotopy types of spaces based on the hypercrossed complexes of Carrasco and Cegarra. | An Alternative Presentation of the Symmetric-Simplicial Category | 13,230 |
We prove a Lefschetz duality result for intersection homology. Usually, this result applies to pseudomanifolds with boundary which are assumed to have a "collared neighborhood of their boundary". Our duality does not need this assumption and is a generalization of the classical one. | A Lefschetz duality intersection homology | 13,231 |
The span of a manifold is its maximum number of linearly independent vector fields. We discuss the question, still unresolved, of whether span(P^m x P^n) always equals span(P^m) + span(P^n). Here P^n denotes real projective space. We use BP-cohomology to obtain new upper bounds for span(P^m x P^n), much stronger than previously known bounds. | Vector fields on RP^m x RP^n | 13,232 |
Dendroidal sets offer a formalism for the study of $\infty$-operads akin to the formalism of $\infty$-categories by means of simplicial sets. We present here an account of the current state of the theory while placing it in the context of the ideas that lead to the conception of dendroidal sets. We briefly illustrate how the added flexibility embodied in $\infty$-operads can be used in the study of $A_{\infty}$-spaces and weak $n$-categories in a way that cannot be realized using strict operads. | From Operads to Dendroidal Sets | 13,233 |
We prove an h-principle with boundary condition for a certain class of topological spaces valued sheaves. The techniques used in the proof come from the study of the homotopy type of the cobordism categories, and they are of simplicial and categorical nature. Applying the main result of this paper to a certain sheaf we find another proof of the Madsen-Weiss theorem, that describes the homotopy type of the classifying space of the cobordism category as the loop space of the Thom space of the complement of the tautological bundle over the Grassmannians. | A relative h-principle via cobordism-like categories | 13,234 |
These are notes, by Z. Fiedorowicz, from lectures given by J. Frank Adams at the University of Chicago in spring of 1973. They give an elegant axiomatic presentation of localization and completion in algebraic topology. The construction of localization and completion functors with respect to an arbitrary generalized homology theory is derived from the axioms by using the Brown representability theorem. These notes were never formally published, due to an apparent flaw in the proof. The relevant representable functors could not be shown to be set-valued, as opposed to class-valued. Subsequent work by A. K. Bousfield established the existence of these functors, using more technical simplicial methods. These functors are now an essential tool in homotopy theory. The notes also contain an addendum devoted to establishing that a certain element in the gamma family of the stable homotopy groups of spheres is nonzero, using Brown-Peterson homology. At that time this was a matter of controversy, as S. Oka and H. Toda claimed to have proved the contrary result. Besides being of historical interest, these notes give a very readable introduction to localization and completion, with minimal prerequisites. A brief epilogue by Z. Fiedorowicz fills the gap in the proof and sketches some follow up history. A foreword and a few additional editorial notes have also been added. | Localisation and Completion with an addendum on the use of
Brown-Peterson homology in stable homotopy | 13,235 |
Shape theory works nice for (Hausdorff) paracompact spaces, but for spaces with no separation axioms, it seems to be quite poor. However, for finite and locally finite spaces their weak homotopy type is rather rich, and is equivalent to the weak homotopy type of finite and locally finite polynedra, respectively. In the paper there is proposed a variant of shape theory called quasi-shape, which suits both paracompact and locally finite spaces, i.e. the quas-shape is isomorphic to the weak homotopy type for locally finite spaces, and is \natural-equivalent to the ordinary shape in the case of paracompact spaces. | Quasi-shape theory of locally finite and paracompact spaces | 13,236 |
The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology of the corresponding moment-angle manifold \mathcal Z_P, and therefore they find numerous applications in combinatorial commutative algebra and toric topology. Here we calculate some bigraded Betti numbers of the type \beta^{-i,2(i+1)} for associahedra, and relate the calculation of the bigraded Betti numbers for truncation polytopes to the topology of their moment-angle manifolds. These two series of simple polytopes provide conjectural extrema for the values of b^{-i,2j}(P) among all simple polytopes P with the fixed dimension and number of vertices. | Bigraded Betti numbers of some simple polytopes | 13,237 |
An explicit chain complex is constructed to calculate the derived functors of destabilization at an odd prime, generalizing constructions of Zarati and of Hung and Sum. The methods are based on the ideas of Singer and Miller and also apply at the prime two. A structural result on the derived functors of destabilization is deduced. | On the derived functors of destabilization at odd primes | 13,238 |
The notion of quadratic self-duality for coalgebras is developed with applications to algebraic structures which arise naturally in algebraic topology, related to the universal Steenrod algebra via an appropriate form of duality. This explains and unifies results of Lomonaco and Singer. | On quadratic coalgebras, duality and the universal Steenrod algebra | 13,239 |
We study functors F from C_f to D where C and D are simplicial model categories and C_f is the full subcategory of C consisting of objects that factor a fixed morphism f from A to B. We define the analogs of Eilenberg and Mac Lane's cross effects functors in this context, and identify explicit adjoint pairs of functors whose associated cotriples are the diagonals of the cross effects. With this, we generalize the cotriple Taylor tower construction of [10] from the setting of functors from pointed categories to abelian categories to that of functors from C_f to D to produce a tower of functors whose n-th term is a degree n functor. We compare this tower to Goodwillie's tower of n-excisive approximations to F found in [8]. When D is a good category of spectra, and F is a functor that commutes with realizations, the towers agree. More generally, for functors that do not commute with realizations, we show that the terms of the towers agree when evaluated at the initial object of C_f. | Cross effects and calculus in an unbased setting | 13,240 |
We give an upper bound on the set of spherical classes in $H_*QX$ when $X = P,S^1$. This is related to the Curtis conjecture on spherical classes in $H_*Q_0S^0$. The results also provide some control over the bordism classes on of immersions when $X$ is a Thom complex. | On spherical classes in $H_*QX$ | 13,241 |
Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a measure of robustness of the homology classes of the filtration of the lower level sets of this function. In this paper we introduce the homology group of filtration as the product of the kernels of the homology maps of the inclusions. This group contains all possible homology classes in all elements of the filtration so that we can later pick the features that lie within the user's choice of the acceptable level of noise. | Homology groups of filtrations | 13,242 |
The non-equivariant topology of Stiefel manifolds has been studied extensively, culminating in a result of Miller demonstrating that a Stiefel manifold splits stably to a wedge of Thom spaces over Grassmannians. Equivariantly, one can consider spaces of isometries between representations as an analogue to Stiefel manifolds. This concept, however, yields a different theory to the non-equivariant case. We first construct a variation on the theory of the functional calculus before studying the homotopy-theoretic properties of this theory. This allows us to construct the main result; a natural tower of G-spectra running down from equivariant isometries which manifests the pieces of the non-equivariant splitting in the form of the homotopy cofibres of the tower. Furthermore, we detail extra topological properties and special cases of this theory, developing explicit expressions covering the tower's geometric and topological structure. We conclude with two detailed conjectures which provide an avenue for future study. Firstly we explore how our theory interacts with the splitting of Miller, proving partial results linking in our work with Miller's and conjecturing even deeper connections. Finally, we begin to calculate the equivariant K-theory of the tower, conjecturing and providing evidence towards the idea that the rich topological structure will be mirrored on the K-theory level by a similarly deep algebraic structure. | The equivariant stable homotopy theory around isometric linear maps | 13,243 |
We show that the category of rational G-spectra for a torus G is Quillen equivalent to an explicit small and practical algebraic model, thereby providing a universal de Rham model for rational G-equivariant cohomology theories. The result builds on the first author's Adams spectral sequence, the second author's functors making rational spectra algebraic. There are several steps, some perhaps of wider interest (1) isotropy separation (replacing the category of G-spectra by modules over a diagram of isotropically simple ring G-spectra) (2) passage to fixed points on ring and module categories (replacing diagrams of ring G-spectra by diagrams of ring spectra) (3) replacing diagrams of ring spectra by diagrams of differential graded algebras (4) rigidity (replacing diagrams of DGAs by diagrams of graded rings). Systematic use of cellularization of model categories is central. | An algebraic model for rational torus-equivariant spectra | 13,244 |
Consider the mod 2 homology spectral sequence associated to a cosimplicial space X. We construct external operations whose target is the spectral sequence associated to E\Sigma_2 \times_{\Sigma_2} (X\times X). If X is a cosimplicial E_\infty-space, we couple these external operations with the structure map E\Sigma_2 \times_{\Sigma_2} (X\times X) \to X to produce internal operations in the spectral sequence. In the sequel we show that they agree with the usual Araki-Kudo operations on the abutment H_*(Tot X). | Operations in the homology spectral sequence of a cosimplicial infinite
loop space | 13,245 |
Let $G$ be a finite group and $k$ be a field of characteristic $p>0$. A cohomology class $\zeta \in H^n(G,k)$ is called productive if it annihilates $\Ext^*_{kG}(L_{\zeta},L_{\zeta})$. We consider the chain complex $\bPz$ of projective $kG$-modules which has the homology of an $(n-1)$-sphere and whose $k$-invariant is $\zeta$ under a certain polarization. We show that $\zeta$ is productive if and only if there is a chain map $\Delta: \bPz \to \bPz \otimes \bPz$ such that $(\id \otimes \epsilon)\Delta\simeq \id$ and $(\epsilon \otimes \id)\Delta \simeq \id$. Using the Postnikov decomposition of $\bPz \otimes \bPz$, we prove that there is a unique obstruction for constructing a chain map $\Delta$ satisfying these properties. Studying this obstruction more closely, we obtain theorems of Carlson and Langer on productive elements. | Productive elements in group cohomology | 13,246 |
We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. Thus we develop model structures for the category of n-polynomial and n-homogeneous functors, along with Quillen pairs relating them. We then classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)-action. This improves upon the classification of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra. | Model Categories for Orthogonal Calculus | 13,247 |
Connected components of $\Map(S^4,B\SU(2))$ are the classifying spaces of gauge groups of principal $\SU(2)$-bundles over $S^4$. Tsukuda [Tsu01] has investigated the homotopy types of connected components of $\Map(S^4,B\SU(2))$. But unfortunately, the proof of Lemma 2.4 in [Tsu01] is not correct for $p=2$. In this paper, we give a complete proof. Moreover, we investigate the further divisibility of $\epsilon_i$ defined in [Tsu01]. In [Tsu], it is shown that divisibility of $\epsilon_i$ have some information about $A_i$-equivalence types of the gauge groups. | A note on homotopy types of connected components of Map(S^4,BSU(2)) | 13,248 |
We show that for any compact Lie group $G$ with identity component $N$ and component group $W=G/N$, the category of free rational $G$-spectra is equivalent to the category of torsion modules over the twisted group ring $H^*(BN)[W]$. This gives an algebraic classification of rational $G$-equivariant cohomology theories on free $G$-spaces and a practical method for calculating the groups of natural transformations between them. This uses the methods of arXiv:1101.2511, and some readers may find the simpler context of the present paper highlights the main thread of the argument. | An algebraic model for free rational G-spectra | 13,249 |
While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author's cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This is used to show that the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface. | Deformation of Singularities and the Homology of Intersection Spaces | 13,250 |
Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to \pi_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}). | Every K(n)-local spectrum is the homotopy fixed points of its Morava
module | 13,251 |
Previously we constructed operations in the mod 2 homology spectral sequence associated to a cosimplicial E-infinity space X. The correct target for this spectral sequence is the homology of Tot X. Noting that in this setting Tot X is an E-infinity space, we show that our operations agree with the usual Araki-Kudo operations in the target. We also prove that the multiplication in the spectral sequence agrees with the multiplication in H_*(Tot X). | Spectral sequence operations converge to Araki-Kudo operations | 13,252 |
We use the spectrum tmf to obtain new nonimmersion results for many real projective spaces RP^n for n as small as 113. The only new ingredient is some new calculations of tmf-cohomology groups. We present an expanded table of nonimmersion results. Our new theorem is new for 17% of the values of n between 2^i and 2^i + 2^14 for i > 14. | Some new nonimmersion results for real projective spaces | 13,253 |
A torus manifold is a closed smooth manifold of dimension $2n$ having an effective smooth $T^n = (S^1)^n$-action with non-empty fixed points. Petrie \cite{petrie:1973} has shown that any homotopy equivalence between a complex projective space $\CP^n$ and a torus manifold homotopy equivalent to $\CP^n$ preserves their Pontrjagin classes. A \emph{generalized Bott manifold} is a closed smooth manifold obtained as the total space of an iterated complex projective space bundles over a point, where each fibration is a projectivization of the Whitney sum of a finite many complex line bundles. For instance, we obtain a product of complex projective spaces if all fibrations are trivial. If each fiber is $\CP^1$, then we call it an (ordinary) \emph{Bott manifold}. In this paper, we investigate the invariance of Pontrjagin classes for torus manifolds whose cohomology ring is isomorphic to that of generalized Bott manifolds. We show that any cohomology ring isomorphism between two torus manifolds whose cohomology ring is isomorphic to that of a product of projective spaces preserves their Pontrjagin classes, which generalizes the Petrie's theorem. In addition, we show that any cohomology ring isomorphism between two torus cohomology Bott manifolds preserves their Pontrjagin classes. As a corollary, there are at most a finite number of torus manifolds homotopy equivalent to either a given product of complex projective space or a given Bott manifold. | Torus actions on cohomology complex generalized Bott manifolds | 13,254 |
Let $M$ be a covariant coefficient system for a finite group $G$. In this paper we analyze several topological abelian groups, some of them new, whose homotopy groups are isomorphic to the Bredon-Illman $G$-equivariant homology theory with coefficients in $M$. We call these groups equivariant Dold-Thom topological groups and we show that they are unique up to homotopy. We use one of the new groups to prove that the Bredon-Illman homology satisfies the infinite-wedge axiom and to make some calculations of the 0th equivariant homology. | Equivariant Dold-Thom topological groups | 13,255 |
Let G be a profinite group, {X_alpha}_alpha a cofiltered diagram of discrete G-spectra, and Z a spectrum with trivial G-action. We show how to define the homotopy fixed point spectrum F(Z, holim_alpha X_alpha)^{hG} and that when G has finite virtual cohomological dimension (vcd), it is equivalent to F(Z, holim_alpha (X_alpha)^{hG}). With these tools, we show that the K(n)-local Spanier-Whitehead dual is always a homotopy fixed point spectrum, a well-known Adams-type spectral sequence is actually a descent spectral sequence, and, for a sufficiently nice k-local profinite G-Galois extension E, with K a closed normal subgroup of G, the equivalence (E^{h_kK})^{h_kG/K} \simeq E^{h_kG} (due to Behrens and the author), where (-)^{h_k(-)} denotes k-local homotopy fixed points, can be upgraded to an equivalence that just uses ordinary (non-local) homotopy fixed points, when G/K has finite vcd. | Function spectra and continuous G-spectra | 13,256 |
If X is a cosimplical $E_{n+1}$ space then Tot(X) is an $E_{n+1}$ space and its mod 2 homology $H_*(Tot(X))$ has Dyer-Lashof and Browder operations. It's natural to ask if the spectral sequence converging to $H_*(Tot(X))$ admits compatible operations. In this paper I give a positive answer to this question. | Homology operations and cosimplicial iterated loop spaces | 13,257 |
We show that the tensor product of two cyclic $A_\infty$-algebras is, in general, not a cyclic $A_\infty$-algebra, but an $A_\infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an $A_\infty$-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $A_\infty$-algebra can be thought of as an $A_\infty$-algebra with homotopy inner products whose higher inner products are trivial. However, the higher inner products on the tensor product of cyclic $A_\infty$-algebras are not necessarily trivial. | Tensor Products of $A_\infty$-algebras with Homotopy Inner Products | 13,258 |
D. K. Biss (Topology and its Applications 124 (2002) 355-371) introduced the topological fundamental group and presented some interesting basic properties of the notion. In this article we intend to extend the above notion to homotopy groups and try to prove some similar basic properties of the topological homotopy groups. We also study more on the topology of the topological homotopy groups in order to find necessary and sufficient conditions for which the topology is discrete. Moreover, we show that studying topological homotopy groups may be more useful than topological fundamental groups. | Topological Homotopy Groups | 13,259 |
It is known that in the integral cohomology of BSO(2m), the square of the Euler class is the same as the Pontryagin class in degree 4m. Given that the Pontryagin classes extend rationally to the cohomology of BSTOP(2m), it is reasonable to ask whether the same relation between the Euler class and the Pontryagin class in degree 4m is still valid in the rational cohomology of BSTOP(2m). In this paper we use smoothing theory and tools from homotopy theory to reformulate the hypothesis, and variants, in a differential topology setting and in a functor calculus setting. | Rational Pontryagin classes and functor calculus | 13,260 |
Using the obstruction theory of Blanc-Dwyer-Goerss, we compute the moduli space of realizations of 2-stage Pi-algebras concentrated in dimensions 1 and n or in dimensions n and n+1. The main technical tools are Postnikov truncation and connected covers of Pi-algebras, and their effect on Quillen cohomology. | Moduli spaces of 2-stage Postnikov systems | 13,261 |
The question of the existence of Universal homotopy commutative and homotopy associative H-spaces (called Abelian H-spaces) is studied. Such a space T(X) would prolong a map from X into an Abelian H-space to a unique H-map from T into X. Examples of such pairs (X,T) are given and conditions are discussed which limit the possible spaces X for which such a T can exist. The Anick spaces are shown not to be universal Abelian H-spaces for the corresponding Moore spaces, but conditions are discussed which could lead to a universal property with respect to a more limited range of targets. | Universal Abelian H-spaces | 13,262 |
This paper is devoted to study some topological properties of the SG subgroup, $\pi_1^{sg}(X,x)$, of the quasitopological fundamental group of a based space $(X,x)$, $\pt$, its topological properties as a subgroup of the topological fundamental group $\pi_1^{\tau}(X,x)$ and its influence on the existence of universal covering of $X$. First, we introduce small generated spaces which have indiscrete topological fundamental groups and also small generated coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of the small generated coverings. Finally, by introducing the notion of semi-locally small generatedness we show that the quasitopological fundamental groups of semi-locally small generated spaces are topological groups. | Topological fundamental groups and small generated coverings | 13,263 |
Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (-1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor. By systematically exploiting strong convergence, we prove several theorems concerning the topological Quillen homology of algebras and modules over operads. These include a theorem relating finiteness properties of topological Quillen homology groups and homotopy groups that can be thought of as a spectral algebra analog of Serre's finiteness theorem for spaces and H.R. Miller's boundedness result for simplicial commutative rings (but in reverse form). We also prove absolute and relative Hurewicz theorems and a corresponding Whitehead theorem for topological Quillen homology. Furthermore, we prove a rigidification theorem, which we use to describe completion with respect to topological Quillen homology (or TQ-completion). The TQ-completion construction can be thought of as a spectral algebra analog of Sullivan's localization and completion of spaces, Bousfield-Kan's completion of spaces with respect to homology, and Carlsson's and Arone-Kankaanrinta's completion and localization of spaces with respect to stable homotopy. We prove analogous results for algebras and left modules over operads in unbounded chain complexes. | Homotopy completion and topological Quillen homology of structured ring
spectra | 13,264 |
The notion of interchange of two multiplicative structures on a topological space is encoded by the tensor product of the two operads parametrizing these structures. Intuitively one might thus expect that the tensor product of an E_m and an E_n operad (which encode the muliplicative structures of m-fold, respectively n-fold loop spaces) ought to be an E_{m+n} operad. However there are easy counterexamples to this naive conjecture. In this paper we show that the tensor product of a cofibrant E_m operad and a cofibrant E_n operad is an E_{m+n} operad. It follows that if A_i are E_{m_i} operads for i=1,2,...,k, then there is an E_{m_1+m_2+...+m_k} operad which maps into their tensor product. | An Additivity Theorem for the Interchange of E_n Structures | 13,265 |
Following an idea of Bendersky-Gitler, we construct an isomorphism between Anderson's and Arone's complexes modelling the chain complex of a map space. This allows us to apply Shipley's convergence theorem to Arone's model. As a corollary, we reduce the problem of homotopy equivalence for certain "toy" spaces to a problem in homological algebra. | On homology of map spaces | 13,266 |
Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? This note provides, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens-Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements, with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group. | Spaces with Noetherian cohomology | 13,267 |
B. Schuster \cite{SCH1} proved that the $mod$ 2 Morava $K$-theory $K(s)^*(BG)$ is evenly generated for all groups $G$ of order 32. For the four groups $G$ with the numbers 38, 39, 40 and 41 in the Hall-Senior list \cite{H}, the ring $K(2)^*(BG)$ has been shown to be generated as a $K(2)^*$-module by transferred Euler classes. In this paper, we show this for arbitrary $s$ and compute the ring structure of $K(s)^*(BG)$. Namely, we show that $K(s)^*(BG)$ is the quotient of a polynomial ring in 6 variables over $K(s)^*(pt)$ by an ideal for which we list explicit generators. | K^*(BG) rings for groups $G=G_{38},...,G_{41}$ of order 32 | 13,268 |
Using sheaf theory, I introduce a continuous theory of persistence for mappings between compact manifolds. In the case both manifolds are orientable, the theory holds for integer coefficients. The sheaf introduced here is stable to homotopic perturbations of the mapping. This stability result has a flavor similar to that of bottleneck stability in persistence. | A Continuous Theory of Persistence for Mappings Between Manifolds | 13,269 |
The tools and arguments developed by Kevin Costello are adapted to families of "Outer Spaces" or spaces of graphs. This allows us to prove a version of Deligne's conjecture: the Harrison homology associated to a homotopy commutative algebra is naturally a module over a particular cobordism category of 3-manifolds. | Topological Field Theories and Harrison Homology | 13,270 |
This paper investigates the relation between Toda brackets and congruences of modular forms. It determines the $f$-invariant of Toda brackets and thereby generalizes the formulas of J.F.\ Adams for the classical $e$-invariant to the chromatic second filtration. | Toda brackets and congruences of modular forms | 13,271 |
This paper is a continuation of our previous work in which we defined the notion of a polytope complex and its $K$-theory. In this paper we produce formulas for the delooping of a simplicial polytope complex and the cofiber of a morphism of simplicial polytope complexes. Along the way we also prove that the (classical and higher) scissors congruence groups of polytopes in a homogeneous $n$-manifold (with sufficient geometric data) are determined by its local properties. | Simplicial Polytope Complexes and Deloopings of $K$-theory | 13,272 |
Let \Sigma = \Sigma _{g,1} be a compact surface of genus g at least 3 with one boundary component, \Gamma its mapping class group and M = H_1(\Sigma , Z) the first integral homology of \Sigma . Using that \Gamma is generated by the Dehn twists in a collection of 2g+1 simple closed curves (Humphries' generators) and simple relations between these twists, we prove that H^1(\Gamma , M) is either trivial or isomorphic to Z. Using Wajnryb's presentation for \Gamma in terms of the Humphries generators we can show that it is not trivial. | A computation of H^1(Γ, H_1(Σ)) | 13,273 |
We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order $n$ but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb. | Alternating sign matrices and domino tilings | 13,274 |
The Dinitz conjecture states that, for each $n$ and for every collection of $n$-element sets $S_{ij}$, an $n\times n$ partial latin square can be found with the $(i,j)$\<th entry taken from $S_{ij}$. The analogous statement for $(n-1)\times n$ rectangles is proven here. The proof uses a recent result by Alon and Tarsi and is given in terms of even and odd orientations of graphs. | The Dinitz problem solved for rectangles | 13,275 |
This report contains expository notes about a function $\vartheta(G)$ that is popularly known as the Lov\'asz number of a graph~$G$. There are many ways to define $\vartheta(G)$, and the surprising variety of different characterizations indicates in itself that $\vartheta(G)$ should be interesting. But the most interesting property of $\vartheta(G)$ is probably the fact that it can be computed efficiently, although it lies ``sandwiched'' between other classic graph numbers whose computation is NP-hard. I~have tried to make these notes self-contained so that they might serve as an elementary introduction to the growing literature on Lov\'asz's fascinating function. | The sandwich theorem | 13,276 |
A generalization of the classical statistics ``maj'' and ``inv'' (the major index and number of inversions) on words is introduced, parameterized by arbitrary graphs on the underlying alphabet. The question of characterizing those graphs that lead to equi-distributed "inv" and "maj" is posed and answered. | The Graphical Major Index | 13,277 |
The number of $n \times n$ matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be $[1!4! >... (3n-2)!]/[n!(n+1)! ... (2n-1)!]$, as conjectured by Mills, Robbins, and Rumsey. | Proof of the Alternating Sign Matrix Conjecture | 13,278 |
On an $r\times (n-r)$ lattice rectangle, we first consider walks that begin at the SW corner, proceed with unit steps in either of the directions E or N, and terminate at the NE corner of the rectangle. For each integer $k$ we ask for $N_k^{n,r}$, the number of {\em ordered\/} pairs of these walks that intersect in exactly $k$ points. The number of points in the intersection of two such walks is defined as the cardinality of the intersection of their two sets of vertices, excluding the initial and terminal vertices. We find two explicit formulas for the numbers $N_k^{n,r}$. Next we note that $N_1^{n,r}= 2 N_0^{n,r}$, i.e., that {\em exactly twice as many pairs of walks have a single intersection as have no intersection\/}. Such a relationship clearly merits a bijective proof, and we supply one. We discuss a number of related results for different assumptions on the two walks. We find the probability that two independent walkers on a given lattice rectangle do not meet. In this situation, the walkers start at the two points $(a,b+x+1)$ and (a+x+1,b)$ in the first quadrant, and walk West or South at each step, except that when a walker reaches the $x$-axis (resp. the $y$-axis) then all future steps are constrained to be South (resp. West) until the origin is reached. We find that if the probability $p(i,j)$ that a step from $(i,j)$ will go West depends only on $i+j$, then the probabilty that the two walkers do not meet until they reach the origin is the same as the probability that a single (unconstrained) walker who starts at $(a, b+x+1)$ and and takes $a+b+x$ steps, finishes at one of the points $(0,1), (-1,2), \ldots, (-x,1+x)$. | Counting pairs of lattice paths by intersections | 13,279 |
Given a set $V$, a subset $S$, and a permutation $\pi$ of $V$, we say that $\pi$ permutes $S$ if $\pi (S) \cap S = \emptyset$. Given a collection $\cS = \{V; S_1,\ldots , S_m\}$, where $S_i \subseteq V ~~(i=1,\ldots ,m)$, we say that $\cS$ is invertible if there is a permutation $\pi$ of $V$ such that $\pi (S_i) \subseteq V-S_i$. In this paper, we present necessary and sufficient conditions for the invertibility of a collection and construct a polynomial algorithm which determines whether a given collection is invertible. For an arbitrary collection, we give a lower bound for the maximum number of sets that can be inverted. Finally, we consider the problem of constructing a collection of sets such that no sub-collection of size three is invertible. Our constructions of such collections come from solutions to the packing problem with unbounded block sizes. We prove several new lower and upper bounds for the packing problem and present a new explicit construction of packing. | Inverting sets and the packing problem | 13,280 |
Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then hypergraph H is invertible iff there exists a permutation pi of V such that for all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility critical if H is not invertible but every hypergraph obtained by removing an edge from H is invertible. The degree of H is d if |{E belongs to H(edges)|x belongs to E}| =< d for each x belongs to V Let i(d) be the maximum number of edges of an invertibility critical hypergraph of degree d. Theorem: i(d) =< (d-1) {2d-1 choose d} + 1. The proof of this result leads to the following covering problem on graphs: Let G be a graph. A family H is subset of (2^{V(G)} is an edge cover of G iff for every edge e of G, there is an E belongs to H(edge set) which includes e. H(edge set) is a minimal edge cover of G iff for H' subset of H, H' is not an edge cover of G. Let b(d) (c(d)) be the maximum cardinality of a minimal edge cover H(edge set) of a complete bipartite graph (complete graph) where H(edge set) has degree d. Theorem: c(d)=< i(d)=<b(d)=< c(d+1) and 3. 2^{d-1} - 2 =< b(d)=< (d-1) {2d-1choose d} +1. The proof of this result uses Sperner theory. The bounds b(d) also arise as bounds on the maximum number of elements in the union of minimal covers of families of sets. | Invertible families of sets of bounded degree | 13,281 |
Let G be a graph with vertices V and edges E. Let F be the union-closed family of sets generated by E. Then F is the family of subsets of V without isolated points. Theorem: There is an edge e belongs to E such that |{U belongs to F | e belongs to U}| =< 1/2|F|. This is equivalent to the following assertion: If H is a union-closed family generated by a family of sets of maximum degree two, then there is an $x$ such that |{U belongs to H | x belongs to U}| > 1/2|H|. This is a special case of the union-closed sets conjecture. To put this result in perspective, a brief overview of research on the union-closed sets conjecture is given. A proof of a strong version of the theorem on graph-generated families of sets is presented. This proof depends on an analysis of the local properties of F and an application of Kleitman's lemma. Much of the proof applies to arbitrary union-closed families and can be used to obtain bounds on |{U belongs to F | e belongs to U}|/|F|. | Graph generated union-closed families of sets | 13,282 |
An expression in the exterior algebra of a Peano space yielding Pappus' Theorem was originally given by Doubilet, Rota, and Stein. Motivated by an identity of Rota, we give an identity in a Grassmann-Cayley algebra of step 3, involving joins and meets alone, which expresses the Theorem of Pappus. | A geometric identity for Pappus' Theorem | 13,283 |
The cycle prefix network is a Cayley coset digraph based on sequences over an alphabet which has been proposed as a vertex symmetric communication network. This network has been shown to have many remarkable communication properties such as a large number of vertices for a given degree and diameter, simple shortest path routing, Hamiltonicity, optimal connectivity, and others. These considerations for designing symmetric and directed interconnection networks are well justified in practice and have been widely recognized in the research community. Among the important properties of a good network, efficient routing is probably one of the most important. In this paper, we further study routing schemes in the cycle prefix network. We confirm an observation first made from computer experiments regarding the diameter change when certain links are removed in the original network, and we completely determine the wide diameter of the network. The wide diameter of a network is now perceived to be even more important than the diameter. We show by construction that the wide diameter of the cycle prefix network is very close to the ordinary diameter. This means that routing in parallel in this network costs little extra time compared to ordinary single path routing. | Restricted routing and wide diameter of the cycle prefix network | 13,284 |
In this paper we show that the set of closure relations on a finite poset P forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closed sets in a convex geometry (in the sense of Edelman and Jamison). We also characterize the modular elements of this lattice and compute its characteristic polynomial. | The lattice of closure relations of a poset | 13,285 |
We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu. | Spanning trees short or small | 13,286 |
Consider a collection of objects, some of which may be `bad', and a test which determines whether or not a given sub-collection contains no bad objects. The non-adaptive pooling (or group testing) problem involves identifying the bad objects using the least number of tests applied in parallel. The `hypergeometric' case occurs when an upper bound on the number of bad objects is known {\em a priori}. Here, practical considerations lead us to impose the additional requirement of {\em a posteriori} confirmation that the bound is satisfied. A generalization of the problem in which occasional errors in the test outcomes can occur is also considered. Optimal solutions to the general problem are shown to be equivalent to maximum-size collections of subsets of a finite set satisfying a union condition which generalizes that considered by Erd\"os \etal \cite{erd}. Lower bounds on the number of tests required are derived when the number of bad objects is believed to be either 1 or 2. Steiner systems are shown to be optimal solutions in some cases. | Optimal pooling designs with error detection | 13,287 |
We described a simple algorithm running in linear time for each fixed constant $k$, that either establishes that the pathwidth of a graph $G$ is greater than $k$, or finds a path-decomposition of $G$ of width at most $O(2^{k})$. This provides a simple proof of the result by Bodlaender that many families of graphs of bounded pathwidth can be recognized in linear time. | A simple linear-time algorithm for finding path-decompositions of small
width | 13,288 |
In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives formulas for the number of plane partitions in each of ten symmetry classes. This paper together with results by Andrews [J. Combin. Theory Ser. A 66 (1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the project of proving all ten formulas. We enumerate cyclically symmetric, self-complementary plane partitions. We first convert plane partitions to tilings of a hexagon in the plane by rhombuses, or equivalently to matchings in a certain planar graph. We can then use the permanent-determinant method or a variant, the Hafnian-Pfaffian method, to obtain the answer as the determinant or Pfaffian of a matrix in each of the ten cases. We row-reduce the resulting matrix in the case under consideration to prove the formula. A similar row-reduction process can be carried out in many of the other cases, and we analyze three other symmetry classes of plane partitions for comparison. | Symmetries of plane partitions and the permanent-determinant method | 13,289 |
In this paper we give graphs with the largest known order for a given degree $\Delta$ and diameter $D$. The graphs are constructed from Moore bipartite graphs by replacement of some vertices by adequate structures. The paper also contains the latest version of the $(\Delta, D)$ table for graphs. | New large graphs with given degree and diameter | 13,290 |
Let F be a family of subsets of {1,2,...,n}. The width-degree of an element x in at least one member of F is the width of the family {U in F | x in U}. If F has maximum width-degree at most k, then F is locally k-wide. Bounds on the size of locally k-wide families of sets are established. If F is locally k-wide and centered (every U in F has an element which does not belong to any member of F incomparable to U), then |F| <= (k+1)(n-k/2); this bound is best possible. Nearly exact bounds, linear in n and k, on the size of locally k-wide families of arcs or segments are determined. If F is any locally k-wide family of sets, then |F| is linearly bounded in n. The proof of this result involves an analysis of the combinatorics of antichains. Let P be a poset and L a semilattice (or an intersection-closed family of sets). The P-size of L is |L^P|. For u in L, the P-density of u is the ratio |[u)^P|/|L^P|. The density of u is given by the [1]-density of u. Let p be the number of filters of P. L has the P-density property iff there is a join-irreducible a in L such that the P-density of a is at most 1/p Which non-trivial semilattices have the P-density property? For P=[1], it has been conjectured that the answer is: "all" (the union-closed sets conjecture). Certain subdirect products of lower-semimodular lattices and, for P=[n], of geometric lattices have the P-density property in a strong sense. This generalizes some previously known results. A fixed lattice has the [n]-density property if n is large enough. The density of a generator U of a union-closed family of sets L containing the empty set is estimated. The estimate depends only on the local properties of L at U. If L is generated by sets of size at most two, then there is a generator U of L with estimated density at most 1/2. | Generalized degrees and densities for families of sets | 13,291 |
Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties. | Notes on the connectivity of Cayley coset digraphs | 13,292 |
The learning complexity of special sets of vertices in graphs is studied in the model(s) of exact learning by (extended) equivalence and membership queries. Polynomial-time learning algorithms are described for vertex covers, independent sets, and dominating sets. The complexity of learning vertex sets of fixed size is also investigated, and it is shown that the k-element vertex covers in a graph can be learned in a number of rounds of interaction that is independent of the size of the graph. Apart from the elegance of these algorithmic problems, the chief motivation is the surprising recently established connection between the important unsolved problem of the learning complexity of CNF (or DNF) formulas and the learning complexity of dominating sets. The complexity of teaching sets of vertices in graphs is also considered. | Algorithms for learning and teaching sets of vertices in graphs | 13,293 |
A method of Proctor [European J. Combin. 5 (1984), no. 4, 331-350] realizes the set of arbitrary plane partitions in a box and the set of symmetric plane partitions as bases of linear representations of Lie groups. We extend this method by realizing transposition and complementation of plane partitions as natural linear transformations of the representations, thereby enumerating symmetric plane partitions, self-complementary plane partitions, and transpose-complement plane partitions in a new way. | Self-complementary plane partitions by Proctor's minuscule method | 13,294 |
An $\{r,s\}$-leaper is a generalized knight that can jump from $(x,y)$ to $(x\pm r,y\pm s)$ or $(x\pm s,y\pm r)$ on a rectangular grid. The graph of an $\{r,s\}$-leaper on an $m\times n$ board is the set of $mn$~vertices $(x,y)$ for $0\leq x<m$ and $0\leq y<n$, with an edge between vertices that are one $\{r,s\}$-leaper move apart. We call $x$ the {\it rank} and $y$ the {\it file} of board position $(x,y)$. George~P. Jelliss raised several interesting questions about these graphs, and established some of their fundamental properties. The purpose of this paper is to characterize when the graphs are connected, for arbitrary~$r$ and~$s$, and to determine the smallest boards with Hamiltonian circuits when $s=r+1$ or $r=1$. | Leaper graphs | 13,295 |
We address the degree-diameter problem for Cayley graphs of Abelian groups (Abelian graphs), both directed and undirected. The problem turns out to be closely related to the problem of finding efficient lattice coverings of Euclidean space by shapes such as octahedra and tetrahedra; we exploit this relationship in both directions. In particular, we find the largest Abelian graphs with 2 generators (dimensions) and a given diameter. (The results for 2 generators are not new; they are given in the literature of distributed loop networks.) We find an undirected Abelian graph with 3 generators and a given diameter which we conjecture to be as large as possible; for the directed case, we obtain partial results, which lead to unusual lattice coverings of 3-space. We discuss the asymptotic behavior of the problem for large numbers of generators. The graphs obtained here are substantially better than traditional toroidal meshes, but, in the simpler undirected cases, retain certain desirable features such as good routing algorithms, easy constructibility, and the ability to host mesh-connected numerical algorithms without any increase in communication times. | The degree-diameter problem for several varieties of Cayley graphs, I:
the Abelian case | 13,296 |
Let $k\ge 1$ be an odd integer, $t=\lfloor {{k+2}\over 4}\rfloor$, and $q$ be a prime power. We construct a bipartite, $q$-regular, edge-transitive graph $C\!D(k,q)$ of order $v \le 2q^{k-t+1}$ and girth $g \ge k+5$. If $e$ is the the number of edges of $C\!D(k,q)$, then $e =\Omega(v^{1+ {1\over {k-t+1}}})$. These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order $v$ and girth at least $g$, $ g\ge 5$, $g \not= 11,12$. For $g\ge 24$, this represents a slight improvement on bounds established by Margulis and Lubotzky, Phillips, Sarnak; for $5\le g\le 23$, $g\not= 11,12$, it improves on or ties existing bounds. | A new series of dense graphs of high girth | 13,297 |
We retrace the recent history of the Umbral Calculus. After studying the classic results concerning polynomial sequences of binomial type, we generalize to a certain type of logarithmic series. Finally, we demonstrate numerous typical examples of our theory. Nous passons en revue ici les resultats recents du calcul ombral. Nous nous interessons tout d'abord aux resultats classique appliqu\'es aux suites de polyn\^omes de type binomial, pius elargions le champ d'\'etude aux series logarithmiques. Enfin nous donnons de nombreaux exemples types d'application de cette th\'eorie. | Recent contributions to the calculus of finite differences: a survey | 13,298 |
Given a universe of discourse $U$, a {\em multiset} can be thought of as a function $M$ from $U$ to the natural numbers ${\bf N}$. In this paper, we define a {\em hybrid set} to be any function from the universe $U$ to the integers ${\bf Z}$. These sets are called hybrid since they contain elements with either a positive or negative multiplicity. Our goal is to use these hybrid sets {\em as if} they were multisets in order to adequately generalize certain combinatorial facts which are true classically only for nonnegative integers. | Getting results with negative thinking | 13,299 |
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