text stringlengths 17 3.36M | source stringlengths 3 333 | __index_level_0__ int64 0 518k |
|---|---|---|
This note explores the interaction between cohomology operations in a generalized cohomology theory and a string topology loop coproduct dual to the Chas--Sullivan loop product. More precisely, we ask for a description for the failure of a given operation to commute with the loop coproduct, and will obtain a satisfactory answer in the case where the operation preserves both sums and products. Examples of such operations include the total Steenrod square in ordinary mod 2 cohomology and the Adams operations in K-theory. | The String Topology Loop Coproduct and Cohomology Operations | 12,900 |
We prove that for any group of the cohomological dimension $n$ the $n$th power of the Berstein class of the group is nontrivial. This allows to prove the following Berstein-Svarc theorem for all $n$: Theorem. For a connected complex $X$ with $\dim X=\cat X=n$, the $n$th power of the Berstein class of $X$ is nontrivial. Previously it was known for $n\ge 3$. We also prove that, for every map $f: M \to N$ of degree $\pm 1$ of closed orientable manifolds, the fundamental group of $N$ is free provided that the fundamental group of $M$ is. | On the Berstein-Svarc Theorem in dimension 2 | 12,901 |
Let $X$ be a finitistic space with its rational cohomology isomorphic to that of the wedge sum $P^2(n)\vee S^{3n} $ or $S^{n} \vee S^{2n}\vee S^{3n}$. We study continuous $\mathbb{S}^1$ actions on $X$ and determine the possible fixed point sets up to rational cohomology depending on whether or not $X$ is totally non-homologous to zero in $X_{\mathbb{S}^1}$ in the Borel fibration $X\hookrightarrow X_{\mathbb{S}^1} \longrightarrow B_{\mathbb{S}^1}$. We also give examples realizing the possible cases. | Fixed points of circle actions on spaces with rational cohomology of
$S^n V S^{2n} V S^{3n}$ or $P^2(n) V S^{3n}$ | 12,902 |
In this paper, we generalize the equivariant homotopy groups or equivalently the Rhodes groups. We establish a short exact sequence relating the generalized Rhodes groups and the generalized Fox homotopy groups and we introduce $\Gamma$-Rhodes groups, where $\Gamma$ admits a certain co-grouplike structure. Evaluation subgroups of $\Gamma$-Rhodes groups are discussed. | A note on generalized equivariant homotopy groups | 12,903 |
For groups of prime order, equivariant stable maps between equivariant representation spheres are investigated using the Borel cohomology Adams spectral sequence. Features of the equivariant stable homotopy category, such as stability and duality, are shown to lift to the category of modules over the associated Steenrod algebra. The dependence on the dimension functions of the representations is clarified. | Equivariant stable stems for prime order groups | 12,904 |
Dold-Thom functors are generalizations of infinite symmetric products, where integer multiplicities of points are replaced by composable elements of a partial abelian monoid. It is well-known that for any connective homology theory, the machinery of $\Gamma$-spaces produces the corresponding linear Dold-Thom functor. In this note we construct such functors directly from spectra by exhibiting a partial monoid corresponding to a spectrum. | Partial monoids and Dold-Thom functors | 12,905 |
For a finite p-group G and a bounded below G-spectrum X of finite type mod p, the G-equivariant Segal conjecture for X asserts that the canonical map X^G --> X^{hG} is a p-adic equivalence. Let C_{p^n} be the cyclic group of order p^n. We show that if the C_p Segal conjecture holds for a C_{p^n} spectrum X, as well as for each of its C_{p^e} geometric fixed points for 0 < e < n, then then C_{p^n} Segal conjecture holds for X. Similar results hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups with suitable finite coefficients. | On cyclic fixed points of spectra | 12,906 |
Let $G$ be a compact connected Lie group and $p : E\to \Sigma A$ be a principal G-bundle with a characteristic map $\alpha : A\to G$, where $A=\Sigma A_{0}$ for some $A_{0}$. Let $\{K_{i}{\to} F_{i-1}{\hookrightarrow} F_{i} \,|\, 1{\le} i {\le} n,\, F_{0}{=} \{\ast\} \; F_{1}{=} \Sigma{K_{1}} \; \text{and}\; F_{n}{\simeq} G \}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=\Sigma{K'_{1}} \subset F_{1}$ with $K'_{1} \subset K_{1}$ which satisfy $F_{i}F'_{1} \subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\operatorname{cat}(X) \le m{+}1$, if firstly the characteristic map $\alpha$ is compressible into $F'_{1}$, secondly the Berstein-Hilton Hopf invariant $H_{1}(\alpha)$ vanishes in $[A, \Omega F'_1{\ast}\Omega F'_1]$ and thirdly $K_{m}$ is a sphere. We apply this to the principal bundle $\mathrm{SO}(9)\hookrightarrow\mathrm{SO}(10)\to S^{9}$ to determine L-S category of $\mathrm{SO}(10)$. | On Lusternik-Schnirelmann category of SO(10) | 12,907 |
The goal of this paper is to ameliorate the sufficients conditions, already established by the first author so that the sum of the numbers of Betti, of 1-connected rational finite CW-complex, is higher than the dimension of his $\mathbb Q$-vectorial space of homotopy, we will present it in two aspects, one algebraic and another geometrical. | The conjecture H: A lower bound of cohomologic dimension for an elliptic
space | 12,908 |
In the mid 1980's, Pete Bousfield and I constructed certain p--local `telescopic' functors Phi_n from spaces to spectra, for each prime p and each positive integer n. These have striking properties that relate the chromatic approach to homotopy theory to infinite loopspace theory: roughly put, the spectrum Phi_n(Z) captures the v_n periodic homotopy of a space Z. Recently there have been a variety of new uses of these functors, suggesting that they have a central role to play in calculations of periodic phenomena. Here I offer a guide to their construction, characterization, application, and computation. | A guide to telescopic functors | 12,909 |
We develop model categories of rational equivariant spectra whose homotopy categories are equivalent to the category of rational equivariant cohomology theories. We prove that given an orthogonal decomposition of the unit in the rational Burnside ring, the model category of rational equivariant spectra decomposes into a product of localisations. We use this result to reprove the classification of rational equivariant cohomology theories for finite groups and to study such cohomology theories for the group O(2). We then concentrate on a split piece of the O(2) case and relate it to rational SO(2) equivariant spectra. | Rational Equivariant Spectra | 12,910 |
We show that topological Quillen homology of algebras and modules over operads in symmetric spectra can be calculated by realizations of simplicial bar constructions. Working with several model category structures, we give a homotopical proof after showing that certain homotopy colimits in algebras and modules over operads can be easily understood. A key result here, which lies at the heart of this paper, is showing that the forgetful functor commutes with certain homotopy colimits. We also prove analogous results for algebras and modules over operads in unbounded chain complexes. | Bar constructions and Quillen homology of modules over operads | 12,911 |
In this paper we describe and continue the study begun by the author, Jones, and Segal, of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a generalization of this, to show that when these moduli spaces are smooth, and are oriented with respect to a generalized cohomology theory E^*, then a Floer E_* -homology theory can be defined. In doing this we describe a functorial viewpoint on how chain complexes can be realized by E -module spectra, generalizing the stable homotopy realization criteria given earlier by the author, Jones, and Segal. Since these moduli spaces, if smooth, will be manifolds with corners, we give a discussion about the appropriate notion of orientations of manifolds with corners. | Floer homotopy theory, realizing chain complexes by module spectra, and
manifolds with corners | 12,912 |
We describe an algebraic structure on chain complexes yielding algebraic models which classify homotopy types of Poincare duality complexes of dimension 4. Generalizing Turaev's fundamental triples of Poincare duality complexes of dimension 3, we introduce fundamental triples for Poincare duality complexes of dimension n > 2 and show that two Poincare duality complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic. As applications we establish a conjecture of Turaev and obtain a criterion for the existence of degree 1 maps between n-dimensional manifolds. | Poincare duality complexes in dimension four | 12,913 |
This is the second in a series of papers. Here we develop here an intersection theory for manifolds equipped with an action of a finite group. As in our previous paper, our approach will be homotopy theoretic, enabling us to circumvent the specter of equivariant transversality. This theory has applications to embedding problems, equivariant fixed point theory and the problem of enumerating the periodic points of a self map of a compact smooth manifold. | Homotopical Intersection Theory, II: equivariance | 12,914 |
To open-closed cobordism surfaces, open-closed string topology associates topological quantum field theory (TQFT) operations, namely string operations, which depend only on homeomorphism types of surfaces and which satisfy the sewing property. We show that most TQFT string operations vanish in open-closed string topology. We describe those open-closed cobordisms with vanishing string operations, and give a short list of open-closed cobordisms with possibly nontrivial string operations. | TQFT string operations in open-closed string topology | 12,915 |
Previously, we showed that most of the open-closed topological quantum field theory (TQFT) string operations vanish including all the higher genus TQFT operations, and we described a small list of genus zero open-closed TQFT string operations which can be nontrivial. In this paper, we consider open-closed string operations associated to open-closed cobordisms homeomorphic to discs. These operations constitute the main part of genus zero string operations, and they include the saddle string operation of two open strings interacting at their internal points. We show not only that disc string operations are independent of their half-pair-of-pants decompositions but also that these disc string operations can be computed by \emph{simultaneous} saddle interactions of incoming open strings at the same point, and they take values in homology classes of constant open strings on some closed orientable submanifolds, which we will precisely determine. We will also discuss a role played by fundamental constant homology classes in open-closed string topology. Our main tools are saddle interaction diagrams and their deformations. | Open-Closed TQFT String Operations for Disc Cobordisms, Simultaneous
Saddle Interactions, and Constant Homology Classes | 12,916 |
For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a consequence of a thorough analysis of the homotopy theory of Moore spaces. Anick further developed this for p>=5 by constructing a homotopy fibration S^2n-1 --> T^2n+1(p^r) --> Loop S^2n+1 whose connecting map is degree p^r on the bottom cell. A much simpler construction of such a fibration for p>=3 was given by Gray and the author using new methods. In this paper the new methods are used to start over, first constructing Anick's fibration for p>=3, and then using it to obtain the exponent result for spheres. | Anick's fibration and the odd primary homotopy exponent of spheres | 12,917 |
Combining results of Wahl, Galatius--Madsen--Tillmann--Weiss and Korkmaz one can identify the homotopy-type of the classifying space of the stable non-orientable mapping class group $N_\infty$ (after plus-construction). At odd primes p, the F_p-homology coincides with that of $Q_0(HP^\infty_+)$, but at the prime 2 the result is less clear. We identify the F_2-homology as a Hopf algebra in terms of the homology of well-known spaces. As an application we tabulate the integral stable homology of $N_\infty$ in degrees up to six. As in the oriented case, not all of these cohomology classes have a geometric interpretation. We determine a polynomial subalgebra of $H^*(N_\infty ; F_2)$ consisting of geometrically-defined characteristic classes. | The homology of the stable non-orientable mapping class group | 12,918 |
Let $G$ be a finite group. For a based $G$-space $X$ and a Mackey functor $M$, a topological Mackey functor $X\widetilde\otimes M$ is constructed, which will be called the stable equivariant abelianization of $X$ with coefficients in $M$. When $X$ is a based $G$-CW complex, $X\widetilde\otimes M$ is shown to be an infinite loop space in the sense of $\mathcal{G}$-spaces. This gives a version of the $RO(G)$-graded equivariant Dold-Thom theorem. Applying a variant of Elmendorf's construction, we get a model for the Eilenberg-Mac Lane spectrum $HM$. The proof uses a structure theorem for Mackey functors and our previous results. | Stable equivariant abelianization, its properties, and applications | 12,919 |
We characterize the 2-line of the p-local Adams-Novikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p > 3. We give a similar characterization of the 1-line, reinterpreting some earlier work of A. Baker and G. Laures. These results are then used to deduce that, for l a prime which generates the p-adic units, the spectrum Q(l) detects the alpha and beta families in the stable stems. | Congruences between modular forms given by the divided beta family in
homotopy theory | 12,920 |
Recently Stephen Theriault and I found an elementary construction of Anick's spaces and proved their main properties(arXiv:0710.1024).In this work the fundamental fibration is decomposed. This is useful in studying maps out of Anick's spaces and will be needed in order to determine it's universal properties. | Decompositions involving Anick's spaces | 12,921 |
A new type of Hopf invariant is described for the fiber of the pinch map from the mapping cone of a map from A to X onto to the suspension of A; this is then used to study the boundary map in the fibration sequence of Cohen, Moore and Neisendorfer in the case that the mapping cone is an odd dimensional Moore space. The components of the boundary map are then shown to be compatible with Hopf invariants and a filtered splitting of the loops on the fiber is obtained. | Filtering the fiber of the pinch map | 12,922 |
Cohen conjectured that for r>=2 there is a space T^2n+1(2^r) and a homotopy fibration sequence Loop^2 S^2n+1 --> S^2n-1 --> T^2n+1(2^r) --> Loop S^2n+1 with the property that the left map composed with the double suspension, Loop^2 S^2n+1 --> S^2n-1 --> Loop^2 S^2n+1, is homotopic to the 2^r-power map. We positively resolve this conjecture when r>=3. Several preliminary results are also proved which are of interest in their own right. | 2-Primary Anick Fibrations | 12,923 |
Not every singular homology class is the push-forward of the fundamental class of some manifold. In the same spirit, one can study the following problem: Which singular homology classes are the push-forward of the fundamental class of a given type of manifolds? In the present article, we show that the fundamental classes of negatively curved manifolds cannot be represented by a non-trivial product of manifolds. This observation sheds some light on the functorial semi-norm on singular homology given by products of compact surfaces. | Fundamental classes of negatively curved manifolds cannot be represented
by products of manifolds | 12,924 |
The category of rational O(2)-equivariant spectra splits as a product of cyclic and dihedral parts. Using the classification of rational G-equivariant spectra for finite groups G, we classify the dihedral part of rational O(2)-equivariant spectra in terms of an algebraic model. | Classifying Dihedral O(2)-Equivariant Spectra | 12,925 |
Let $f:G_{n,k}\longrightarrow G_{m,l}$ be any continuous map between any two distinct complex Grassmann manifolds of the same dimension where the target is not the complex projective space. We show that, for any given $k,l$, the degree of $f$ is zero provided that $m,n$ are sufficiently large. If the degree of $f$ is $\pm 1$, we show that $(m,l)=(n,k)$ and $f$ is a homotopy equivalence. Also, we prove that the image under $f^*$ of elements of a set of algebra generators of $H^*(G_{m,l};\mathbb{Q})$ is determined upto a sign, $\pm$, if the degree of $f$ is non-zero. Our proofs cover the case of quaternionic Grassmann manifolds as well. | Degrees of maps between Grassmann manifolds | 12,926 |
We consider the category of presheaves of Gamma-spaces, or equivalently, of Gamma-objects in simplicial presheaves. Our main result is the construction of stable model structures on this category parametrised by local model structures on simplicial presheaves. If a local model structure on simplicial presheaves is monoidal, the corresponding stable model structure on presheaves of Gamma-spaces is monoidal and satisfies the monoid axiom. This allows us to lift the stable model structures to categories of algebras and modules over commutative algebras. | Homotopy theory of presheaves of Gamma-spaces | 12,927 |
Let $G$ be a group acting continuously on a space $X$ and let $X/G$ be its orbit space. Determining the topological or cohomological type of the orbit space $X/G$ is a classical problem in the theory of transformation groups. In this paper, we consider this problem for cohomology lens spaces. Let $X$ be a finitistic space having the mod 2 cohomology algebra of the lens space $L_p^{2m-1}(q_1,...,q_m)$. Then we classify completely the possible mod 2 cohomology algebra of orbit spaces of arbitrary free involutions on $X$. We also give examples of spaces realizing the possible cohomology algebras. In the end, we give an application of our results to non-existence of $\mathbb{Z}_2$-equivariant map $\mathbb{S}^n \to X$. | Cohomology algebra of orbit spaces of free involutions on lens spaces | 12,928 |
Twisted diagrams are "diagrams" with components in different categories. Structure maps are defined using auxiliary data which consists of functors relating the various categories to each other. Prime examples of the construction are spectra (in the sense of homotopy theory) and quasi-coherent sheaves on schemes. We develop the basic theory of twisted diagrams, and establish various model structures (which are well-known in special cases). We also introduce a notion of homotopy sheaves, a collection of local data which is compatible up to weak equivalence, and study basic properties of such objects. These objects occur in nature; for example, the notion of an Omega-spectrum fits into this framework. The main purpose of the paper is to provide a convenient reference for model structures on twisted diagrams, and for the language of sheaves and homotopy sheaves as defined here. | Twisted diagrams and homotopy sheaves | 12,929 |
We give a self contained introduction to A$_\infty$-algebras, A$_\infty$-bimodules and maps between them. The case of A$_\infty$-bimodule-map between $A$ and its dual space $A^{*}$, which we call $\infty$-inner-product, will be investigated in detail. In particular, we describe the graph complex associated to $\infty$-inner-product. In a later paper, we show how $\infty$-inner-products can be used to model the string topology BV-algebra on the free loop space of a Poincar\'e duality space. | Infinity inner products on A-infinity algebras | 12,930 |
A topological group is constructed which is homotopy equivalent to the pointed loop space of a path-connected Riemannian manifold $M$ and which is given in terms of "composable small geodesics" on $M$. This model is analogous to J. Milnor's free group construction \cite{Milnor} which provides a model for the pointed loop space of a connected simplicial complex. Related function spaces are constructed from "composable small geodesics" which provide models for the free loop space of $M$ as well as the space of continuous maps from a surface to $M$. | On "small geodesics" and free loop spaces | 12,931 |
Suppose that G=S^1 acts freely on a finitistic space X whose mod p cohomology ring isomorphic to that of a lens space L^{2m-1}(p;q_1,...,q_m). In this paper, we determine the mod p cohomology ring of the orbit space X/G. If the characteristic class \alpha\belongs H^2(X/G;Z_p) of the S^1-bundle S--> X--> X/G is nonzero, then mod p ndex of the action is deined to be the largest integer n such that \alpha^n is nonzero. We also show that the mod p index of a free action of S^1 on a lens space L^(2m-1)(p;q_1,...,q_m) is p-1, provided that \alpha is nonzero. | Cohomology algebra of the orbit space of free circle group actions on
lens spaces | 12,932 |
We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence. | Topological nonrealization results via the Goodwillie tower approach to
iterated loopspace homology | 12,933 |
The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of n distinct points in Euclidean m-space for all m>1$ and n>1; the answer was previously known in the cases m=2 and m odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity. | Topological complexity of configuration spaces | 12,934 |
We examine the proof of a classical localization theorem of Bousfield and Friedlander and we remove the assumption that the underlying model category be right proper. The key to the argument is a lemma about factoring in morphisms in the arrow category of a model category. | Note on a theorem of Bousfield and Friedlander | 12,935 |
The theory of secondary chomology operations leads to a conjecture concerning the algebra of higher cohomology operations in general. This conjecture is discussed here in detail and its connection with homotopy groups of spheres and the Adams spectral sequence is described. | A conjecture on homotopy groups of spheres, details on the algebra of
higher cohomology operations | 12,936 |
In Computer Vision the ability to recognize objects in the presence of occlusions is a necessary requirement for any shape representation method. In this paper we investigate how the size function of a shape changes when a portion of the shape is occluded by another shape. More precisely, considering a set $X=A\cup B$ and a measuring function $\phi$ on $X$, we establish a condition so that $\ell_{(X,\phi)=\ell_{(A,\phi|_A)}+\ell_{(B,\phi|_B)}-\ell_{(A\cap B,\phi|_{A\cap B})}$. The main tool we use is the Mayer-Vietoris sequence of \v{C}ech homology groups. This result allows us to prove that size functions are able to detect partial matching between shapes by showing a common subset of cornerpoints. | Cech homology for shape recognition in the presence of occlusions | 12,937 |
The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called `algebraic' because they originate from abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the source categories are usually very `non-additive' before passing to homotopy classes of morphisms. Because of their origin I refer to these examples as `topological triangulated categories'. In these extended talk notes I explain some systematic differences between these two kinds of triangulated categories. There are certain properties -- defined entirely in terms of the triangulated structure -- which hold in all algebraic examples, but which fail in some topological ones. These differences are all torsion phenomena, and rationally there is no difference between algebraic and topological triangulated categories. | Algebraic versus topological triangulated categories | 12,938 |
Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using derived functors and the symmetric bar construction of Fiedorowicz. The symmetric homology of group rings is related to stable homotopy theory. Two chain complexes are constructed that compute symmetric homology, as well as two spectral sequences. In the setup of the second spectral sequence, a complex isomorphic to the suspension of the cycle-free chessboard complex of Vrecica and Zivaljevic appears. Homology operations are defined on the symmetric homology groups over Z/p, p a prime. Finally, an explicit partial resolution is presented, permitting the computation of the zeroth and first symmetric homology groups of finite-dimensional algebras. | On the Symmetric Homology of Algebras | 12,939 |
We generalize a semi-norm for the Alexander polynomial of a connected, compact, oriented 3-manifold on its first cohomology group to a semi-norm for an arbitrary Laurent polynomial f on the dual vector space to the space of exponents of f. We determine a decomposition formula for this Laurent norm; an expression for the Laurent norm for f in terms of the Laurent norms for each of the irreducible factors of f. For an n-variable polynomial f, we introduce a space of m \leq n essential variables which determine the reduced Laurent norm unit ball; a convex polyhedron of the same dimension m as the Newton polyhedron of f. In the space spanned by the essential variables, the Laurent semi-norm for polynomials with at least two terms is shown to be a norm. | The Laurent norm | 12,940 |
We show that the Alexander and Thurston norms are the same for all irreducible Eisenbud-Neumann graph links in homology 3-spheres. These are the links obtained by splicing Seifert links in homology 3-spheres together along tori. By combining this result with previous results, we prove that the two norms coincide for all links in S^3 if either of the following two conditions are met; the link is a graph link, so that the JSJ decomposition of its complement in S^3 is made up of pieces which are all Seifert-fibered, or the link is alternating and not a (2,n)-torus link, so that the JSJ decomposition of its complement in S^3 is made up of pieces which are all hyperbolic. We use the E-N obstructions to fibrations for graph links together with the Thurston cone theorem on link fibrations to deduce that every facet of the reduced Thurston norm unit ball of a graph link is a fibered facet. | Alexander and Thurston norms of graph links | 12,941 |
Let E be a k-local profinite G-Galois extension of an E_infty-ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete G-spectrum. Also, we prove that if E is a profaithful k-local profinite extension which satisfies certain extra conditions, then the forward direction of Rognes's Galois correspondence extends to the profinite setting. We show the function spectrum F_A((E^hH)_k, (E^hK)_k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]])^hK)_k where H and K are closed subgroups of G. Applications to Morava E-theory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points. | The homotopy fixed point spectra of profinite Galois extensions | 12,942 |
Let X be a finitistic space with non-trivial cohomology groups H^in(X;Z)=Z with generators v_i, where i = 0, 1, 2, 3. We say that X has cohomology type (a, b) if v_1^2 = av_2 and v_1v_2 = bv_3 . In this note, we determine the mod 2 cohomology ring of the orbit space X/G of a free action of G = Z_2 on X, where both a and b are even. In this case, we observed that there is no equivariant map S^m --> X for m > 3n, where S^m has the antipodal action. Moreover, it is shown that G can not act freely on space X which is of cohomology type (a, b) where a is odd and b is even. We also obtain the mod 2 cohomology ring of the orbit space X/G of free action of G = S^1 on the space X of type (0, b). | Cohomology algebra of the orbit space of some free actions on spaces of
cohomology type (a, b) | 12,943 |
We show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its "deformation representation ring," analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices. | The Bott cofiber sequence in deformation K-theory and simultaneous
similarity in U(n) | 12,944 |
For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps, and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even torsion lens spaces and complex projective spaces are discussed. | Symmetric topological complexity of projective and lens spaces | 12,945 |
In previous work, the authors have each introduced methods for studying the 2-line of the p-local Adams-Novikov spectral sequence in terms of the arithmetic of modular forms. We give the precise relationship between the congruences of modular forms introduced by the first author with the Q-spectrum and the f-invariant of the second author. This relationship enables us to refine the target group of the f-invariant in a way which makes it more manageable for computations. | beta-family congruences and the f-invariant | 12,946 |
We give a conjectural description for the kernel of the map assigning to each finite $\mathbb Z_p$-free $G\times\mathbb Z_p$-set its rational permutation module where G is a finite p-group. We prove that this conjecture is true when G is an elementary abelian p-group or a cyclic p-group. | The Relative Burnside Kernel - The Elementary Abelian Case | 12,947 |
We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\widehat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\widehat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\widehat{\mathcal O}$ in terms of derivations and module maps. As an application we construct a homotopy inner product over the commutative operad on the cochains of any Poincar\'e duality space. | Homotopy Inner Products for Cyclic Operads | 12,948 |
We show that in closed string topology and in open-closed string topology with one $D$-brane, higher genus stable string operations are trivial. This is a consequence of Harer's stability theorem and related stability results on the homology of mapping class groups of surfaces with boundaries. In fact, this vanishing result is a special case of a general result which applies to all homological conformal field theories with a property that in the associated topological quantum field theories, the string operations associated to genus one cobordisms with one or two boundaries vanish. In closed string topology, the base manifold can be either finite dimensional, or infinite dimensional with finite dimensional cohomology for its based loop space. The above vanishing result is based on the triviality of string operations associated to homology classes of mapping class groups which are in the image of stabilizing maps. | Stable string operations are trivial | 12,949 |
This paper is an English translation of chapter nine of the book "Adams spectral sequence and stable homotopy groups of spheres" by Jinkun Lin (in Chinese)(Sciences Press, Beijing 2007). In this paper, a sequence of new indecomposable families in the stable homotopy groups of spheres such as h_0h_n,h_nb_n,h_0h_nh_m,h_0(h_mb_{n-1}-b_{m-1}h_n) families was detected. The proof of all the detections was stated in this paper in based on several published papers by the author. | English translation of chapter 9 of the book "Adams spectral sequence
and stable homotopy groups of spheres" by Jijkun Lin (In Chinese)(Sciences
Press, Beijing 2007) | 12,950 |
We give a new solution of the "homotopy periods" problem, as highlighted by Sullivan, which places explicit geometrically meaningful formulae first dating back to Whitehead in the context of Quillen's formalism for rational homotopy theory and Koszul-Moore duality. Geometrically, we show that homotopy groups are rationally given by "generalized linking/intersection invariants" of cochain data. Moreover, we give a method for determining when two maps from $S^n$ to $X$ are homotopic after allowing for multiplication by some integer. For applications, we investigate wedges of spheres and homogeneous spaces (where homotopy is given by classical linking numbers), and configuration spaces (where homotopy is given by generalized linking numbers); also we propose a generalization of the Hopf invariant one question. | Lie coalgebras and rational homotopy theory II: Hopf invariants | 12,951 |
This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the conference "New Topological Contexts for Galois Theory and Algebraic Geometry." | An overview of abelian varieties in homotopy theory | 12,952 |
In this paper we study the topology of cobordism categories of manifolds with corners. Specifically, if {Cob}_{d,<k>} is the category whose objets are a fixed dimension d, with corners of codimension less than or equal to k, then we identify the homotopy type of the classifying space B{Cob}_{d,<k>} as the zero space of a homotopy colimit of certain diagram of Thom spectra. We also identify the homotopy type of the corresponding cobordism category when extra tangential structure is assumed on the manifolds. These results generalize the results of Galatius, Madsen, Tillmann and Weiss, and the proofs are an adaptation of the their methods. As an application we describe the homotopy type of the category of open and closed strings with a background space X, as well as its higher dimensional analogues. This generalizes work of Baas-Cohen-Ramirez and Hanbury. | Cobordism categories of manifolds with corners | 12,953 |
We propose a generalization of Sullivan's de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan's theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions. | Rational homotopy theory and differential graded category | 12,954 |
We study the topology of spaces related to Kac-Moody groups. Given a split Kac-Moody group over the complex numbers, let K denote the unitary form with maximal torus T having normalizer N(T). In this article we study the cohomology of the flag manifold K/T, as a module over the Nil-Hecke ring, as well as the (co)homology of K as a Hopf algebra. In particular, if F is a field of positive characteristic, we show that H_*(K,F) is a finitely generated algebra, and that H^*(K,F) is finitely generated only if K is a compact Lie group . We also study the stable homotopy type of the classifying space BK and show that it is a retract of the classifying space BN(T). We illustrate our results with the example of rank two Kac-Moody groups. | On the Topology of Kac-Moody groups | 12,955 |
In this article we describe relations of the topology of closed 1-forms to the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CW- complexes and show that many properties of the group theoretic version have analogous statements. In particular we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik- Schnirelmann category of a closed 1-form and to the existence of a non- singular closed 1-form in a given cohomology class on a high-dimensional closed manifold. | Closed 1-Forms in Topology and Geometric Group Theory | 12,956 |
Let p, ell be distinct primes and let q be a power of p. Let G be a connected compact Lie group. We show that there exists an integer b such that the mod ell cohomology of the classifying space of a finite Chevalley group G(F_q) is isomorphic to the mod ell cohomology of the classifying space of the loop group LG for q=p^{ab}, a>0. | Finite Chevalley groups and loop groups | 12,957 |
We define and study a homological version of Sullivan's rational de Rham complex for simplicial sets. This new functor can be generalised to simplicial symmetric spectra and in that context it has excellent categorical properties which promise to make a number of interesting applications much more straightforward. | Chains on suspension spectra | 12,958 |
In this paper we describe two ways on which cofibred categories give rise to bisimplicial sets. The "fibred nerve" is a natural extension of Segal's classical nerve of a category, and it constitutes an alternative simplicial description of the homotopy type of the total category. If the fibration is splitting, then one can construct the "cleaved nerve", a smaller variant which emerges from a distinguished closed cleavage. We interpret some classical theorems by Thomason and Quillen in terms of our constructions, and use the fibred and cleaved nerve to establish new results on homotopy and homology of small categories. | On the homotopy type of a cofibred category | 12,959 |
We establish certain conditions which imply that a map $f:X\to Y$ of topological spaces is null homotopic when the induced integral cohomology homomorphism is trivial; one of them is: $H^*(X)$ and $\pi_*(Y)$ have no torsion and $H^*(Y)$ is polynomial. | On the homotopy classification of maps | 12,960 |
We study topological spaces with a distinguished set of paths, called directed paths. Since these directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category, called the fundamental category. We define models of the fundamental category, such as the fundamental bipartite graph, and minimal extremal models which are shown to generalize the fundamental group. In addition, we prove van Kampen theorems for subcategories, retracts, and models of the fundamental category. | Models and van Kampen theorems for directed homotopy theory | 12,961 |
Given a simply connected space $X$ with the cohomology $H^*(X;{\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(\Omega X;{\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\mathbb Z}_2).$ As a consequence we obtain that $H^*(\Omega X;{\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\ast}(X;{\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel. | The loop cohomology of a space with the polynomial cohomology algebra | 12,962 |
We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units gl(A). To a map of spectra f: b -> bgl(A), we associate a commutative A-algebra Thom spectrum Mf, which admits a commutative A-algebra map to R if and only if b -> bgl(A) -> bgl(R) is null. If A is an associative ring spectrum, then to a map of spaces f: B -> BGL(A) we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B -> BGL(A) -> BGL(R) is null. We also note that BGL(A) classifies the twists of A-theory. We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory. | Units of ring spectra and Thom spectra | 12,963 |
Let $\pi: E \to B$ be a fiber bundle with fiber having the mod 2 cohomology algebra of a real or a complex projective space and let $\pi^{'}: E^{'} \to B$ be vector bundle such that $\mathbb{Z}_2$ acts fiber preserving and freely on $E$ and $E^{'}-0$, where 0 stands for the zero section of the bundle $\pi^{'}:E^{'} \to B$. For a fiber preserving $\mathbb{Z}_2$-equivariant map $f:E \to E^{'}$, we estimate the cohomological dimension of the zero set $Z_f = \{x \in E | f(x)= 0\}.$ As an application, we also estimate the cohomological dimension of the $\mathbb{Z}_2$-coincidence set $A_f=\{x \in E | f(x) = f(T(x)) \}$ of a fiber preserving map $f:E \to E^{'}$. | Parametrized Borsuk-Ulam problem for projective space bundles | 12,964 |
We show that there is an essentially unique S-algebra structure on the Morava K-theory spectrum K(n), while K(n) has uncountably many MU or \hE{n}-algebra structures. Here \hE{n} is the K(n)-localized Johnson-Wilson spectrum. To prove this we set up a spectral sequence computing the homotopy groups of the moduli space of A-infinity structures on a spectrum, and use the theory of S-algebra k-invariants for connective S-algebras due to Dugger and Shipley to show that all the uniqueness obstructions are hit by differentials. | Uniqueness of Morava K-theory | 12,965 |
We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an E-infinity operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E-infinity algebras. We use this effective definition to prove that the n-fold bar complex B^n(A) admits an extension to categories of algebras over E_n-operads. Then we prove that the n-fold bar complex determines the homology theory associated to a category of algebras over E_n-operads. For n infinite, we obtain an isomorphism between the homology of an infinite bar construction and the usual Gamma-homology with trivial coefficients. | Iterated bar complexes of E-infinity algebras and homology theories | 12,966 |
Numerably contractible spaces play an important role in the theory of homotopy pushouts and pullbacks. The corresponding results imply that a number of well known weak homotopy equivalences are genuine ones if numerably contractible spaces are involved. In this paper we give a first systematic investigation of numerably contractible spaces. We list the elementary properties of the category of these spaces. We then study simplicial objects in this category. In particular, we show that the topological realization functor preserves fibration sequences if the base is path-connected and numerably contractible in each dimension. Consequently, the loop space functor commutes with realization up to homotopy. We give simple conditions which assure that free algebras over a topological operad are numerably contractible. | Numerably Contractible Spaces | 12,967 |
We describe the constructible derived category of sheaves on the $n$-sphere, stratified in a point and its complement, as a dg module category of a formal dg algebra. We prove formality by exploring two different methods: As a combinatorial approach, we reformulate the problem in terms of representations of quivers and prove formality for the 2-sphere, for coefficients in a principal ideal domain. We give a suitable generalization of this formality result for the 2-sphere stratified in several points and their complement. As a geometric approach, we give a description of the underlying dg algebra in terms of differential forms, which allows us to prove formality for $n$-spheres, for real or complex coefficients. | Formality of the constructible derived category for spheres: A
combinatorial and a geometric approach | 12,968 |
In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These homotopy groups have already been calculated by Shimomura. The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura. An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the K(2)-localization of the spectrum TMF of topological modular forms and related spectra. Even more, the Adams-Novikov differentials for L_{K(2)}V(0) can be read off from those for TMF. | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited | 12,969 |
We study the mod-p cohomology of the group Out(F_n) of outer automorphisms of the free group F_n in the case n=2(p-1) which is the smallest n for which the p-rank of this group is 2. For p=3 we give a complete computation, at least above the virtual cohomological dimension of Out(F_4) (which is 5). More precisley, we calculate the equivariant cohomology of the p-singular part of outer space for p=3. For a general prime p>3 we give a recursive description in terms of the mod-p cohomology of Aut(F_k) for k less or equal to p-1. In this case we use the Out(F_{2(p-1)})-equivariant cohomology of the poset of elementary abelian p-subgroups of Out(F_n). | On the mod - p cohomology of Out(F_{2(p-1)} | 12,970 |
We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space L(BX). This leads to explicit calculations of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p and HZ. | Topological Hochschild homology of Thom spectra and the free loop space | 12,971 |
In this paper we analyze the higher topological Hochschild homology of commutative Thom S-algebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the homotopy orbits of the torus action on iterated topological Hochschild homology and we describe the relationship to topological Andre-Quillen homology. | Higher topological Hochschild homology of Thom spectra | 12,972 |
We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E_\infty classifying map X -> BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E_\infty ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result THH(Mf) \htp Mf \sma BX_+ which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X -> BG is only a homotopy commutative A_\infty map, provided that the induced multiplication on Mf extends to an E_\infty ring structure; this permits us to recover Bokstedt's calculation of THH(HZ). | THH of Thom spectra that are E_\infty ring spectra | 12,973 |
This is the Ph.D. dissertation of the author. The project has been motivated by the conjecture that the Hopkins-Miller tmf spectrum can be described in terms of `spaces' of conformal field theories. In this dissertation, spaces of field theories are constructed as classifying spaces of categories whose objects are certain types of field theories. If such a category has a symmetric monoidal structure and its components form a group, by work of Segal, its classifying space is an infinite loop space and defines a cohomology theory. This has been carried out for two classes of field theories: (i) For each integer n, there is a category SEFT_n whose objects are the Stolz-Teichner (1|1)-dimensional super Euclidean field theories of degree n. It is proved that the classifying space |SEFT_n| represents degree-n K or KO cohomology, depending on the coefficients of the field theories. (ii) For each integer n, there is a category AFT_n whose objects are a kind of (2|1)-dimensional field theories called `annular field theories,' defined using supergeometric versions of circles and annuli only. It is proved that the classifying space |AFT_n| represents the degree-n elliptic cohomology associated with the Tate curve. To the author's knowledge, this is the first time the definitions of low-dimensional supersymmetric field theories are given in full detail. | Supersymmetric field theories and cohomology | 12,974 |
The primary goal of this paper is to find a homotopy theoretic approximation to moduli spaces of holomorphic maps Riemann surfaces into complex projective space. There is a similar treatment of a partial compactification of these moduli spaces of consisting of irreducible stable maps in the sense of Gromov-Witten theory. The arguments follow those from a paper of G. Segal on the topology of the space of rational functions. | Homological Stability among Moduli Spaces of Holomorphic Curves in
Complex Projective Space | 12,975 |
In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex structures along with a holomorphic map to a target complex manifold. A general notion of "geometric structure" is defined using sheaf theoretic constructions. Our main theorem is the identification of the homotopy type of such cobordism categories in terms of certain Thom spectra. This extends work of Galatius-Madsen-Tillmann-Weiss who identify the homotopy type of cobordism categories of manifolds with fiberwise structures on their tangent bundles. Interpretations of the main theorem are discussed which have relevance to topological field theories, moduli spaces of geometric structures, and h-principles. Applications of the main theorem to various examples of interest in geometry, particularly holomorphic curves, are elaborated upon. | Geometric Cobordism Categories | 12,976 |
The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified Treumann, based on unpublished ideas of MacPherson, constructed an `exit category' (in the terminology of this paper, the `fundamental category') which classifies constructible sheaves, equivalently stratified etale covers. This paper generalises this construction to homotopically stratified sets, in addition showing that the fundamental category dually classifies constructible cosheaves, equivalently stratified branched covers. The more general setting has several advantages. It allows us to remove a technical `tameness' condition which appears in Treumann's work; to show that the fundamental groupoid can be recovered by inverting all morphisms and, perhaps most importantly, to reduce computations to the two stratum case. This provides an approach to computing the fundamental category in terms of homotopy groups of strata and homotopy links. We apply these techniques to compute the fundamental category of symmetric products of R^2, stratified by collisions. Two appendices explain the close relations respectively between filtered and pre-ordered spaces and between cosheaves and branched covers (technically locally-connected uniquely-complete spreads). | The fundamental category of a stratified space | 12,977 |
The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over $\mathbb{Z}$ is recovered. | A volume form on the Khovanov invariant | 12,978 |
If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit S forms a commutative ring. An idempotent e of this ring will split the homotopy category. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to the product of C localised at the object eS and C localised at the object (1-e)S. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is. | Splitting Monoidal Stable Model Categories | 12,979 |
We give a new proof that for a finite group G, the category of rational G-equivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model. | Classifying Rational G-Spectra for Finite G | 12,980 |
The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a "local preorder" encoding control flow. In the case where time does not loop, the "locally preordered" state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a "locally monotone" covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes. | Covering space theory for directed topology | 12,981 |
The elements of the ring of bidegree (0,0) additive unstable operations in complex K-theory can be described explicitly as certain infinite sums of Adams operations. Here we show how to make sense of the same expressions for complex cobordism MU, thus identifying the "Adams subring" of the corresponding ring of cobordism operations. We prove that the Adams subring is the centre of the ring of bidegree (0,0) additive unstable cobordism operations. For an odd prime p, the analogous result in the p-local split setting is also proved. | Infinite sums of additive unstable Adams operations and cobordism | 12,982 |
We study Whitehead products in the rational homotopy groups of a general component of a function space. For the component of any based map f: X \to Y, in either the based or free function space, our main results express the Whitehead product directly in terms of the Quillen minimal model of f. These results follow from a purely algebraic development in the setting of chain complexes of derivations of differential graded Lie algebras, which is of interest in its own right. We apply the results to study the Whitehead length of function space components. | Whitehead products in function spaces: Quillen model formulae | 12,983 |
We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the "building complex" associated to level 3 structures at the prime 2. Finally, we note the existence of a number of connective models of the spectrum TMF(Gamma_0(3)). | Topological Modular Forms of Level 3 | 12,984 |
We are interested in the relationship between the virtual cohomological dimension (or vcd) of a discrete group Gamma and the smallest possible dimension of a model for the classifying space of Gamma relative to its family of virtually cyclic subgroups. In this paper we construct a model for the virtually cyclic classifying space of the Heisenberg group. This model has dimension 3, which equals the vcd of the Heisenberg group. We also prove that there exists no model of dimension less than 3. | The Virtually Cyclic Classifying Space of the Heisenberg Group | 12,985 |
We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object X homotopy equivalent to an algebra A over a cofibrant prop P inherits a P-algebra structure so that X defines a model of A in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of A-infinity algebras. | Props in model categories and homotopy invariance of structures | 12,986 |
The homotopy type and homotopy groups of some spectra TAF of topological automorphic forms associated to a unitary similitude group GU of type (1,1) are explicitly described in quasi-split cases. The spectrum TAF is shown to be closely related to the spectrum TMF in these cases, and homotopy groups of some of these spectra are explicitly computed. | Topological automorphic forms on U(1,1) | 12,987 |
We consider discontinuous operations of a group $G$ on a contractible $n$-dimensional manifold $X$. Let $E$ be a finite dimensional representation of $G$ over a field $k$ of characteristics 0. Let $\mathcal{E}$ be the sheaf on the quotient space $Y=G \setminus X$ associated to $E$. Let $H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$ be the image in $H^{\bullet}(Y;\mathcal{E})$ of the cohomology with compact support. In the cases where both $H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$ and $H^{\bullet}_{\textbf{!}}(Y;\mathcal{E}^*)$ ($\mathcal{E}^*$ being the the sheaf associated to the representation dual to $E$) are finite dimensional, we establish a non-degenerate duality between $H^{m}_{\textbf{!}}(Y;\mathcal{E})$ and $H^{n-m}_{\textbf{!}}(Y;\mathcal{E}^{\ast})$. We also show that this duality is compatible with Hecke operators. | Another point in homological algebra: Duality for discontinuous group
actions | 12,988 |
The notion of the \emph{homotopy type} of a topological stack has been around in the literature for some time. The basic idea is that an atlas $X \to \mathfrak{X}$ of a stack determines a topological groupoid $\mathbb{X}$ with object space $X$. The homotopy type of $\mathfrak{X}$ should be the classifying space $B \mathbb{X}$. The choice of an atlas is not part of the data of a stack and hence it is not immediately clear why this construction of a homotopy type is well-defined, let alone functorial. The purpose of this note is to give an elementary construction of such a homotopy-type functor. | The homotopy type of a topological stack | 12,989 |
We provide an axiomatic framework for the study of smooth extensions of generalized cohomology theories. Our main results are about the uniqeness of smooth extensions, and the identification of the flat theory with the R/Z-theory. In particular, we show that there is a unique smooth extension of K-theory and of MU-cobordism with a unique multiplication, and that the flat theory in these cases is naturally isomorphic to the homotopy theorist's version of the cohomology theory with R/Z-coefficients. For this we only require a small set of natural compatibility conditions. | Uniqueness of smooth extensions of generalized cohomology theories | 12,990 |
Based on a Whitehead-type characterization of the sectional category we develop the notion of weak sectional category. This is a new lower bound of the sectional category, which is inspired by the notion of weak category in the sense of Berstein-Hilton. We establish several properties and inequalities, including the fact that the weak sectional category is a better lower bound for the sectional category than the classical one given by the nilpotency of the kernel of the induced map in cohomology. Finally, we apply our results in the study of the topological complexity in the sense of Farber. | Weak sectional category | 12,991 |
The purpose of this paper is twofold. First, we review applications of the bar duality of operads to the construction of explicit cofibrant replacements in categories of algebras over an operad. In view toward applications, we check that the constructions of the bar duality work properly for algebras over operads in unbounded differential graded modules over a ring. In a second part, we use the operadic cobar construction to define explicit cyclinder objects in the category of operads. Then we apply this construction to prove that certain homotopy morphisms of algebras over operads are equivalent to left homotopies in the model category of operads. | Operadic bar constructions, cylinder objects, and homotopy morphisms of
algebras over operads | 12,992 |
In this paper we introduce an open-closed cobordism category with maps to a background space. We identify the classifying space of this category for certain classes of background space. The key ingredient is the homology stability of mapping class groups with twisted coefficients. | An open-closed cobordism category with background space | 12,993 |
Let G be a discrete group which acts properly and isometrically on a complete CAT(0)-space X. Consider an integer d with d=1 or d greater or equal to 3 such that the topological dimension of X is bounded by d. We show the existence of a G-CW-model E_fin(G) for the classifying space for proper G-actions with dim(E_fin(G)) less or equal to d. Provided that the action is also cocompact, we prove the existence of a G-CW-model E_vcyc(G) for the classifying space of the family of virtually cyclic subgroups such that dim(E_vcyc(G)) is less or equal to d+1. | On the classifying space of the family of finite and of virtually cyclic
subgroups for CAT(0)-groups | 12,994 |
The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to stable homotopy theory via $HS_*(k[\Gamma]) \cong H_*(\Omega\Omega^{\infty} S^{\infty}(B\Gamma); k)$. Two chain complexes that compute $HS_*(A)$ are constructed, both making use of a symmetric monoidal category $\Delta S_+$ containing $\Delta S$. Two spectral sequences are found that aid in computing symmetric homology. The second spectral sequence is defined in terms of a family of complexes, $Sym^{(p)}_*$. $Sym^{(p)}$ is isomorphic to the suspension of the cycle-free chessboard complex $\Omega_{p+1}$ of Vre\'{c}ica and \v{Z}ivaljevi\'{c}, and so recent results on the connectivity of $\Omega_n$ imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the $k\Sigma_{p+1}$--module structure of $Sym^{(p)}$ are devloped. A partial resolution is found that allows computation of $HS_1(A)$ for finite-dimensional $A$ and some concrete computations are included. | Symmetric Homology of Algebras | 12,995 |
If G is a group acting properly by semisimple isometries on a proper CAT(0) space X, then we build models for the classifying spaces E_{vc} and E_{fbc} under the additional assumption that the action of G has a well-behaved collection of axes in X. (This hypothesis is described in the paper.) We conjecture that the latter hypothesis is satisfied in a large range of cases. Our classifying spaces resemble those created by Connolly, Fehrman, and Hartglass for crystallographic groups G. | Constructions of E_{vc} and E_{fbc} for groups acting on CAT(0) spaces | 12,996 |
A planar compactum with connected complement can be an embedded in a cellular continuum by attaching a null sequence of arcs. Two based maps f and g from a planar Peano continuum X to a planar set Y are homotopic iff f and g induce the same homomorphism between fundamental groups. | Embedding planar compacta in planar continua with application: homotopic
maps between planar Peano continua are characterized by the fundamental group
homomorphism | 12,997 |
We study the Gray index of phantom maps, which is a numerical invariant of phantom maps. It is conjectured that the only phantom map with infinite Gray index between finite-type spaces is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of phantom maps are restricted to simply connected finite complexes. As an application of the counter example we show that $\SNT^{\infty}(X)$ can be non-trivial for some space $X$ of finite type. | On the Gray index conjecture for phantom maps | 12,998 |
We prove a congruence criterion for the algebraic theory of power operations in Morava E-theory, analogous to Wilkerson's congruence criterion for torsion free lambda-rings. In addition, we provide a geometric description of this congruence criterion, in terms of sheaves on the moduli problem of deformations of formal groups and Frobenius isogenies. | The congruence criterion for power operations in Morava E-theory | 12,999 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.