text stringlengths 17 3.36M | source stringlengths 3 333 | __index_level_0__ int64 0 518k |
|---|---|---|
As a first goal, it is explained why Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P^m only for m < 16. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential, but critical, high-order obstructions in the corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an n-dimensional Euclidean embedding of P^m. A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis' BP-approach to the immersion problem of P^m. A form of the Euler class viewpoint is applied to show TC^S(P^3) = 5, as well as to suggest a few higher dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber's work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber's lead, this concept is connected to the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P^3. | Symmetric topological complexity as the first obstruction in
Goodwillie's Euclidean embedding tower for real projective spaces | 13,100 |
In this paper we constructs a new nontrivial family in the stable homotopy groups of spheres $\pi_{p^nq+2pq+q-3}S$ which is of order $p$ and is represented by $k_0h_{n} \in Ext_A^{3,p^nq+2pq+q}(\mathbb{Z}_p,\mathbb{Z}_p)$ in the Adams spectral sequence, where $p\geq 5$ is an odd prime, $n\geq 3$ and $q=2(p-1)$. In the course of the proof, a new family of homotopy elements in $\pi_{\ast}V(1)$ which is represented by $\beta_{\ast}{i^{\prime}}_{\ast}i_{\ast}({h}_n)\in Ext_A^{2,p^nq+(p+1)q+1}(H^{\ast}V(1),\mathbb{Z}_p)$ in the Adams sequence is detected. | Detection of some elements in the stable homotopy groups of spheres | 13,101 |
Let ($\Omega^{\ast}(M), d$) be the de Rham cochain complex for a smooth compact closed manifolds $M$ of dimension $n$. For an odd-degree closed form $H$, there are a twisted de Rham cochain complex $(\Omega^{\ast}(M), d+H_\wedge)$ and its associated twisted de Rham cohomology $H^*(M,H)$. We show that there exists a spectral sequence $\{E^{p, q}_r, d_r\}$ derived from the filtration $F_p(\Omega^{\ast}(M))=\bigoplus_{i\geq p}\Omega^i(M)$ of $\Omega^{\ast}(M)$, which converges to the twisted de Rham cohomology $H^*(M,H)$. We also show that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper. | On a spectral sequence for twisted cohomologies | 13,102 |
Manfred Stelzer has pointed out that part of Corollary 4.5 of our paper "Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology" [Alg. Geo. Top. 8 (2008), 2109--2129] was not sufficiently proved, and, indeed, is likely incorrect as stated. This necessitates a little more argument to finish the proof of the main theorem of the original paper. The statement of this theorem, and all the examples, remain unchanged. | Correction to ""Topological nonrealization results via the Goodwillie
tower approach to iterated loopspace homology" | 13,103 |
We provide a model of the String group as a central extension of finite-dimensional 2-groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more na\"ive 2-category of Lie groupoids, smooth functors, and smooth natural transformations. In particular this notion of smooth 2-group subsumes the notion of Lie 2-group introduced by Baez-Lauda. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by G. Segal, and our String 2-group is a special case of such extensions. There is a nerve construction which can be applied to these 2-groups to obtain a simplicial manifold, allowing comparison with with the model of A. Henriques. The geometric realization is an $A_\infty$-space, and in the case of our model, has the correct homotopy type of String(n). Unlike all previous models our construction takes place entirely within the framework of finite dimensional manifolds and Lie groupoids. Moreover within this context our model is characterized by a strong uniqueness result. It is a unique central extension of Spin(n). | Central Extensions of Smooth 2-Groups and a Finite-Dimensional String
2-Group | 13,104 |
We construct a cobordism group for embedded graphs in two different ways, first by using sequences of two basic operations, called "fusion" and "fission", which in terms of cobordisms correspond to the basic cobordisms obtained by attaching or removing a 1-handle, and the other one by using the concept of a 2-complex surface with boundary is the union of these knots. A discussion given to the question of extending Khovanov homology from links to embedded graphs, by using the Kauffman topological invariant of embedded graphs by associating family of links and knots to a such graph by using some local replacements at each vertex in the graph. This new concept of Khovanov homology of an embedded graph constructed to be the sum of the Khovanov homologies of all the links and knots. | Khovano Homology and Embedded Graphs | 13,105 |
Our aim in this paper is to give a geometric description of the cup product in negative degrees of Tate cohomology of a finite group with integral coefficients. By duality it corresponds to a product in the integral homology of $BG$: {\[H_{n}(BG,\mathbb{Z})\otimes H_{m}(BG,\mathbb{Z})\rightarrow H_{n+m+1}(BG,\mathbb{Z})\]} for $n,m>0$. We describe this product as join of cycles, which explains the shift in dimensions. Our motivation came from the product defined by Kreck using stratifold homology. We then prove that for finite groups the cup product in negative Tate cohomology and the Kreck product coincide. The Kreck product also applies to the case where $G$ is a compact Lie group (with an additional dimension shift). | On the Product in Negative Tate Cohomology for Finite Groups | 13,106 |
In this paper we study the spaces of $q$-tuples of points in a Euclidean space, without $k$-wise coincidences (configuration-like spaces). A transitive group action by permuting these points is considered, and some new upper bounds on the genus (in the sense of Krasnosel'skii--Schwarz and Clapp--Puppe) for this action are given. Some theorems of Cohen--Lusk type for coincidence points of continuous maps to Euclidean spaces are deduced. | Configuration-like spaces and coincidences of maps on orbits | 13,107 |
The Curtis conjecture predicts that the only spherical classes in $H_*(Q_0S^0;\Z/2)$ are the Hopf invariant one and the Kervaire invariant one elements. We consider Sullivan's decomposition $$Q_0S^0=J\times\cok J$$ where $J$ is the fibre of $\psi^q-1$ ($q=3$ at the prime $2$) and observe that the Curtis conjecture holds when we restrict to $J$. We then use the Bott periodicity and the $J$-homomorphism $SO\to Q_0S^0$ to define some generators in $H_*(Q_0S^0;\Z/p)$, when $p$ is any prime, and determine the type of subalgebras that they generate. For $p=2$ we determine spherical classes in $H_*(\Omega^k_0J;\Z/2)$. We determine truncated subalgebras inside $H_*(Q_0S^{-k};\Z/2)$. Applying the machinery of the Eilenberg-Moore spectral sequence we define classes that are not in the image of by the $J$-homomorphism. We shall make some partial observations on the algebraic structure of $H_*(\Omega^k_0\cok J;\Z/2)$. Finally, we shall make some comments on the problem in the case equivariant $J$-homomorphisms. | On the Bott periodicity, $J$-homomorphisms, transfer maps and
$H_*Q_0S^{-n}$ | 13,108 |
We determine the integral cohomology ring of the homogeneous space E_8/T^1E_7 by the Borel presentation and a method due to Toda. Then using the Gysin exact sequence associated with the circle bundle S^1 -> E_8/E_7 -> E_8/T^1E_7, we also determine the integral cohomology of E_8/E_7. | The integral cohomology ring of E_8/T^1E_7 | 13,109 |
We construct a stable model structure on profinite symmetric spectra with a continuous action of an arbitrary profinite group. This provides a natural framework for a new construction of homotopy fixed point spectra and of homotopy fixed point spectral sequences for the action of the extended Morava stabilizer group on Lubin-Tate spectra. These continuous homotopy fixed points are canonically equivalent to the homotopy fixed points of Devinatz and Hopkins but have a drastically simplified construction. | Continuous homotopy fixed points for Lubin-Tate spectra | 13,110 |
Starting with a Z-graded superconnection on a graded vector bundle over a smooth manifold M, we show how Chen's iterated integration of such a superconnection over smooth simplices in M gives an A-infinity functor if and only if the superconnection is flat. If the graded bundle is trivial, this gives a twisting cochain. Very similar results were obtained by K.T. Chen using similar methods. This paper is intended to explain this from scratch beginning with the definition and basic properties of a connection and ending with an exposition of Chen's "formal connections" and a brief discussion of how this is related to higher Reidemeister torsion. | Iterated integrals of superconnections | 13,111 |
We give a complete description of the integral cohomology ring of the flag manifold E_8/T, where E_8 denotes the compact exceptional Lie group of rank 8 and T its maximal torus, by the method due to Borel and Toda. This completes the computation of the integral cohomology rings of the flag manifolds for all compact connected simple Lie groups. | The integral cohomology ring of E_8/T | 13,112 |
We introduce some compact orbifolds on which there is a certain finite group action having a simple convex polytope as the orbit space. We compute the orbifold fundamental group and homology groups of these orbifolds. We calculate the cohomology rings of these orbifolds when the dimension of the orbifold is even. These orbifolds are intimately related to the notion of small cover. | Some small orbifolds over polytopes | 13,113 |
Lannes' T-functor is used to give a construction of the Singer functor R_1 on the category U of unstable modules over the Steenrod algebra A. This leads to a direct proof that the composite functor Fix R_1 is naturally equivalent to the identity. Further properties of the functors R_1 are deduced, especially when applied to reduced and nilclosed unstable modules. | On the Singer functor R_1 and the functor Fix | 13,114 |
The cohomological rigidity problem for toric manifolds asks whether the cohomology ring of a toric manifold determines the topological type of the manifold. In this paper, we consider the problem with the class of one-twist Bott manifolds to get an affirmative answer to the problem. We also generalize the result to quasitoric manifolds. In doing so, we show that the twist number of a Bott manifold is well-defined and is equal to the cohomological complexity of the cohomology ring of the manifold. We also show that any cohomology Bott manifold is homeomorphic to a Bott manifold. All these results are also generalized to the case with $\mathbb Z_{(2)}$-coefficients, where $\mathbb Z_{(2)}$ is the localized ring at 2. | Properties of Bott manifolds and cohomological rigidity | 13,115 |
We use the orientation underlying the Hirzebruch genus of level three to map the beta family at the prime p=2 into the ring of divided congruences. This procedure, which may be thought of as the elliptic greek letter beta construction, yields the f-invariants of this family. | The beta family at the prime two and modular forms of level three | 13,116 |
The purpose of this paper is to study generalizations of Gamma-homology in the context of operads. Good homology theories are associated to operads under appropriate cofibrancy hypotheses, but this requirement is not satisfied by usual operads outside the characteristic zero context. In that case, the idea is to pick a cofibrant replacement Q of the given operad P. We can apply to P-algebras the homology theory associated to Q in order to define a suitable homology theory on the category of P-algebras. We make explicit a small complex to compute this homology when the operad P is binary and Koszul. In the case of the commutative operad P=Com, we retrieve the complex introduced by Robinson for the Gamma-homology of commutative algebras. | Gamma-homology of algebras over an operad | 13,117 |
We prove that the Goodwillie tower of a weak equivalence preserving functor from spaces to spectra can be expressed in terms of the tower for stable mapping spaces. Our proof is motivated by interpreting the functors P_n and D_n as pseudo-differential operators which suggests certain `integral' presentations based on a derived Yoneda embedding. These models allow one to extend computational tools available for the tower of stable mapping spaces. As an application we give a classical expression for the derivative over the basepoint. | Models For The Maclaurin Tower Of A Simplicial Functor Via A Derived
Yoneda Embedding | 13,118 |
It is well known that very special $\Gamma$-spaces and grouplike $\E_\infty$ spaces both model connective spectra. Both these models have equivariant analogues. Shimakawa defined the category of equivariant $\Gamma$-spaces and showed that special equivariant $\Gamma$-spaces determine positive equivariant spectra. Costenoble and Waner showed that grouplike equivariant $\E_\infty$-spaces determine connective equivariant spectra. We show that with suitable model category structures the category of equivariant $\Gamma$-spaces is Quillen equivalent to the category of equivariant $\E_\infty$ spaces. We define the units of equivariant ring spectra in terms of equivariant $\Gamma$-spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum. | Units of equivariant ring spectra | 13,119 |
We prove that an additive track category with strong coproducts is equivalent to the category of pseudomodels for the algebraic theory of $\nil_2$ groups. This generalizes the classical statement that the category of models for the algebraic theory of abelian groups is equivalent to the category of abelian groups. Dual statements are also considered. | The $Γ$-structure of an additive track category | 13,120 |
A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $\C P^1$-bundle starting with a point, where each $\C P^1$-bundle is the projectivization of a Whitney sum of two complex line bundles. A \emph{$\Q$-trivial Bott manifold} of dimension $2n$ is a Bott manifold whose cohomology ring is isomorphic to that of $(\CP^1)^n$ with $\Q$-coefficients. We find all diffeomorphism types of $\Q$-trivial Bott manifolds and show that they are distinguished by their cohomology rings with $\Z$-coefficients. As a consequence, we see that the number of diffeomorphism classes in $\Q$-trivial Bott manifolds of dimension $2n$ is equal to the number of partitions of $n$. We even show that any cohomology ring isomorphism between two $\Q$-trivial Bott manifolds is induced by a diffeomorphism. | Classification of Q-trivial Bott manifolds | 13,121 |
We combine Lurie's generalization of the Hopkins-Miller theorem with work of Zink-Lau on displays to give a functorial construction of even-periodic commutative ring spectra, concentrated in chromatic layers 2 and above, associated to certain n by n invertible matrices with coefficients in Witt rings. This is applied to examples related to Lubin-Tate and Johnson-Wilson spectra. We also give a Hopf algebroid presentation of the moduli of p-divisible groups of height greater than or equal to 2. | Structured ring spectra and displays | 13,122 |
Let $X$ be a finitistic space having the mod 2 cohomology algebra of the product of two projective spaces. We study free involutions on $X$ and determine the possible mod 2 cohomology algebra of orbit space of any free involution, using the Leray spectral sequence associated to the Borel fibration $X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}$. We also give an application of our result to show that if $X$ has the mod 2 cohomology algebra of the product of two real projective spaces (respectively complex projective spaces), then there does not exist any $\mathbb{Z}_2$-equivariant map from $\mathbb{S}^k \to X$ for $k \geq 2$ (respectively $k \geq 3$), where $\mathbb{S}^k$ is equipped with the antipodal involution. | Orbit spaces of free involutions on the product of two projective spaces | 13,123 |
We prove structural theorems for computing the completion of a G-spectrum at the augmentation ideal of the Burnside ring of a finite group G. First we show that a G-spectrum can be replaced by a spectrum obtained by allowing only isotropy groups of prime power order without changing the homotopy type of the completion. We then show that this completion can be computed as a homotopy colimit of completions of spectra obtained by further restricting isotropy to one prime at a time, and that these completions can be computed in terms of completion at a prime. As an application, we show that the spectrum of stable maps from BG to the classifying space of a compact Lie group K splits non-equivariantly as a wedge sum of p-completed suspension spectra of classifying spaces of certain subquotients of the product of G and K. In particular this describes the dual of BG. | Completion of $G$-spectra and stable maps between classifying spaces | 13,124 |
By studying the representation theory of a certain infinite $p$-group and using the generalised characters of Hopkins, Kuhn and Ravenel we find useful ways of understanding the rational Morava $E$-theory of the classifying spaces of general linear groups over finite fields. Making use of the well understood theory of formal group laws we establish more subtle results integrally, building on relevant work of Tanabe. In particular, we study in detail the cases where the group has dimension less than or equal to the prime $p$ at which the $E$-theory is localised. | The Morava E-theories of finite general linear groups | 13,125 |
We survey the topology which led to the original bar and cobar constructions, for both associative algebras and coalgebras and for Lie algebras and commutative coalgebras. These constructions are often viewed as part of the larger theory of Koszul duality of operads, so this survey is meant to offer an historical perspective on the most prominent cases of that theory. We also explain recent work which shows that Hopf/linking invariants for homotopy are at the heart of the duality between commutative algebras and Lie coalgebras. | Koszul duality in algebraic topology - an historical perspective | 13,126 |
We gather conditions on a class H of continuous maps of topological spaces that allow a reasonable theory of fibrations up to an equivalence (a map from this class) which we call H-fibrations. The weak homotopy equivalences recover quasifibrations and homology equivalences yield homology fibrations. We study local H-fibrations that behave nicely with respect to homotopy colimits together with universal H-fibrations that behave nicely with respect to pullbacks. We then proceed to classify H-fibrations up to a natural notion of equivalence. | Fibrations up to an equivalence, homotopy colimits and pullbacks | 13,127 |
We give a classification of minimal algebras generated in degree 1, defined over any field $\bk$ of characteristic different from 2, up to dimension 6. This recovers the classification of nilpotent Lie algebras over $\bk$ up to dimension 6. In the case of a field $\bk$ of characteristic zero, we obtain the classification of nilmanifolds of dimension less than or equal to 6, up to $\bk$-homotopy type. Finally, we determine which rational homotopy types of such nilmanifolds carry a symplectic structure. | Classification of Minimal Algebras over any Field up to Dimension 6 | 13,128 |
A simple counterexample is presented to a proposition which is used in the arguments given by P. M. Akhmet'ev in his work on the Hopf invariant and Kervaire invariant. The counterexample makes use of the $K$-theory of the quotient of the 7-sphere by the quaternion group of order 8. | $K$-theory of $S^7/Q_8$ and a counterexample to a result of P.M.
Akhmet'ev | 13,129 |
In this article, we investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups. | Module Structure on Lie Powers and Natural Coalgebra-Split Sub Hopf
Algebras of Tensor Algebras | 13,130 |
We present a new technique for analyzing the v_0-Bockstein spectral sequence studied by Shimomura and Yabe. Employing this technique, we derive a conceptually simpler presentation of the homotopy groups of the E(2)-local sphere for p > 3. We identify and correct some errors in the original Shimomura-Yabe calculation. We deduce the related K(2)-local homotopy groups, and discuss their manifestation of Gross-Hopkins duality. | The homotopy groups of the E(2) local sphere at p > 3 revisited | 13,131 |
Let $X$ be a co-$H$-space of $(p-1)$-cell complex with all cells in even dimensions. Then the loop space $\Omega X$ admits a retract $\bar A^{\min}(X)$ that is the evaluation of the functor $\bar A^{\min}$ on $X$. In this paper, we determine the homology $H_*(\bar A^{\min}(X))$ and give the $\EHP$ sequence for the spaces $\bar A^{\min}(X)$. | The Functor $A^{\min}$ for $(p-1)$-cell Complexes and $\EHP$ Sequences | 13,132 |
Looking at the cartesian product of a topological space with itself, a natural map to be considered on that object is the involution that interchanges the coordinates, i.e. that maps (x,y) to (y,x). The so-called halfsquaring construction, now also called "symmetric squaring construction", in Cech homology with Z/2-coefficients was introduced in [arXiv:0709.1774] as a map from the k-th Cech homology group of a space X to the 2k-th Cech homology group of X \times X divided by the above mentioned involution. It turns out to be a crucial construction in the proof of a Borsuk-Ulam-type theorem. In this thesis, a generalization of this construction, which is here called "Dachabbildung", to Cech homology with integer coefficients is given for even dimensions k, which is proven to equally satisfy the useful properties of the original construction. Detailed proofs are provided which could also supersede the somewhat sketchy exposition of [arXiv:0709.1774] for the case of Z/2-coefficients. As a preparation for the presented results, two possibilities of defining Cech homology are closely examined and shown to be ismorphic. | Die Dachabbildung in ganzzahliger Cech-Homologie | 13,133 |
We investigate {\it Gottlieb map}s, which are maps $f:E\to B$ that induce the maps between the Gottlieb groups $\pi_n (f)|_{G_n(E)}:G_n(E)\to G_n(B)$ for all $n$, from a rational homotopy theory point of view.We will define the obstruction group $O(f)$ to be a Gottlieb map and a numerical invariant $o(f)$. It naturally deduces a relative splitting of $E$ in certain cases. We also illustrate several rational examples of Gottlieb maps and non-Gottlieb maps by using derivation arguments in Sullivan models. | A rational obstruction to be a Gottlieb map | 13,134 |
Let $f:X\to Y$ be a pointed map between connected CW-complexes. As a generalization of the evaluation subgroup $G_*(Y,X;f)$, we will define the {\it relaxed evaluation subgroup} ${\mathcal G}_*(Y,X;f)$ in the homotopy group $\pi_*(Y)$ of $Y$, which is identified with ${\rm Im} \pi_*(\tilde{ev})$ for the evaluation map $\tilde{ev} :map(X,Y;f)\times X\to Y$ given by $\tilde{ev} (h,x)=h(x)$. Especially we see by using Sullivan model in rational homotopy theory for the rationalized map $f_{\Q}$ that ${\mathcal G}_*(Y_{\Q},X_{\Q};f_{\Q})=\pi_*(Y)\otimes \Q$ if the map $f$ induces an injection of rational homotopy groups. Also we compare it with more relaxed subgroups by several rationalized examples. | A relaxed evaluation subgroup | 13,135 |
We study the cohomology of spaces of string links and braids in $\mathbb{R}^n$ for $n\geq 3$ using configuration space integrals. For $n>3$, these integrals give a chain map from certain diagram complexes to the deRham algebra of differential forms on these spaces. For $n=3$, they produce all finite type invariants of string links and braids. | On the cohomology of spaces of links and braids via configuration space
integrals | 13,136 |
We explore an approach to twisted generalized cohomology from the point of view of stable homotopy theory and quasicategory theory provided by arXiv:0810.4535. We explain the relationship to the twisted K-theory provided by Fredholm bundles. We show how our approach allows us to twist elliptic cohomology by degree four classes, and more generally by maps to the four-stage Postnikov system BO<0...4>. We also discuss Poincare duality and umkehr maps in this setting. | Twists of K-theory and TMF | 13,137 |
An \'etale homotopy type $T(X, z)$ associated to any pointed locally fibrant connected simplicial sheaf $(X, z)$ on a pointed locally connected small Grothendieck site $(\mc{C}, x)$ is studied. It is shown that this type $T(X, z)$ specializes to the \'etale homotopy type of Artin-Mazur for pointed connected schemes $X$, that it is invariant up to pro-isomorphism under pointed local weak equivalences (but see \cite{Schmidt1} for an earlier proof), and that it recovers abelian and nonabelian sheaf cohomology of $X$ with constant coefficients. This type $T(X, z)$ is compared to the \'etale homotopy type $T_b(X, z)$ constructed by means of diagonals of pointed bisimplicial hypercovers of $x = (X, z)$ in terms of the associated categories of cocycles, and it is shown that there are bijections \pi_0 H_{\hyp}(x, y) \cong \pi_0 H_{\bihyp}(x, y) at the level of path components for any locally fibrant target object $y$. This quickly leads to natural pro-isomorphisms $T(X, z) \cong T_b(X, z)$ in $\Ho{\sSet_\ast}$. By consequence one immediately establishes the fact that $T_b(X, z)$ is invariant up to pro-isomorphism under pointed local weak equivalences. Analogous statements for the unpointed versions of these types also follow. | Etale Homotopy Types and Bisimplicial Hypercovers | 13,138 |
Let $p$ be an odd prime, and fix integers $m$ and $n$ such that $0<m<n\leq (p-1)(p-2)$. We give a $p$-local homotopy decomposition for the loop space of the complex Stiefel manifold $W_{n,m}$. Similar decompositions are given for the loop space of the real and symplectic Stiefel manifolds. As an application of these decompositions, we compute upper bounds for the $p$-exponent of $W_{n,m}$. Upper bounds for $p$-exponents in the stable range $2m<n$ and $0<m\leq (p-1)(p-2)$ are computed as well. | Homotopy Decompositions of Looped Stiefel manifolds, and their Exponents | 13,139 |
We describe the projectives in the category of functors from a graded poset to abelian groups. Based on this description we define a related condition, pseudo-projectivity, and we prove that this condition is enough for the vanishing of the derived direct limits. We apply this result to deduce a generalized version of a theorem of Whitehead for the pushout. The dual results for inverse limits are also considered. We present two methods to compute integral cohomology of posets, a local one a and a global one. The local method is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes, simplex-like posets and for Quillen's complex of a finite group. The global method is related to Morse Theory. | Homological Algebra on Graded Posets | 13,140 |
As an application of the upper triangular technology method of (V.P. Snaith: {\em Stable homotopy -- around the Arf-Kervaire invariant}; Birkh\"{a}user Progress on Math. Series vol. 273 (April 2009)) it is shown that there do not exist stable homotopy classes of $ {\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with the Hopf map to $ {\mathbb RP}^{\infty}$ followed by the Kahn-Priddy map gives an element in the stable homotopy of spheres of Arf-Kervaire invariant one. | Non-factorisation of Arf-Kervaire classes through ${\mathbb RP}^{\infty}
\wedge {\mathbb RP}^{\infty}$ | 13,141 |
Let $R\subseteq \Bbb Q$ be a subring of the rationals and let $p$ be the least prime (if none, $p=\infty $) which is not invertible in $R.$ For an $R$-local $r$-connected $CW$-complex $X$ of dimension $\leq \min(r+2p-3,rp-1), r\geq 1, $ a complete homotopy invariant is constructed in terms of the loop space homology $H_*(\Omega X).$ This allows us to classify all such $R$-local spaces up to homotopy with a fixed loop space homology. | On the homotopy classification of spaces by the fixed loop space
homology | 13,142 |
This paper concerns our earlier conjecture about the equivalence of a derived completion construction applied to the representation spectrum of the absolute Galois group of a geometric field is equivalent to the algebraic K-theory of the field. We prove that the p-adic version of the derived completion construction can be interpreted as the K-theory of a certain pro-scheme over an algebraically closed field contained within the field. In order to make this construction, we use a special property of absolute Galois groups of geometric field, namely that of total torsion freeness, and the paper also contains some poperties of the representation theory of such groups. | K-theoretic descent and a motivic Atiyah-Segal theorem | 13,143 |
Let X and Y be finite nilpotent CW complexes with dimension of X less than the connectivity of Y. Generalizing results of Vigu\'e-Poirrier and Yamaguchi, we prove that the mapping space Map(X,Y) is rationally formal if and only if Y has the rational homotopy type of a finite product of odd dimensional spheres. | Rational formality of mapping spaces | 13,144 |
The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We give an answer by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes. | Crossed modules and the homotopy 2-type of a free loop space | 13,145 |
An E_1 (or A-infinity) ring spectrum R has a derived category of modules D_R. An E_2 structure on R endows D_R with a monoidal product. An E_3 structure on R endows the monoidal product with a braiding. If the E_3 structure extends to an E_4 structure then the braided monoidal product is symmetric monoidal. | The smash product for derived categories in stable homotopy theory | 13,146 |
We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with simple and twisted coefficients---of the dihedral group of order 8 (in the case of unordered configurations) and the elementary abelian 2-group of rank 2 (in the case of ordered configurations). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimension 2^i + d for d=0,1,2. | The integral cohomology groups of configuration spaces of pairs of
points in real projective spaces | 13,147 |
We give natural descriptions of the homology and cohomology algebras of regular quotient ring spectra of even E-infinity ring spectra. We show that the homology is a Clifford algebra with respect to a certain bilinear form naturally associated to the quotient ring spectrum F. To identify the cohomology algebra, we first determine the derivations of F and then prove that the cohomology is isomorphic to the exterior algebra on the module of derivations. We treat the example of the Morava K-theories in detail. | Clifford algebras from quotient ring spectra | 13,148 |
We construct a free and transitive action of the group of bilinear forms Bil(I/I^2[1]) on the set of R-products on F, a regular quotient of an E-infinity ring spectrum R with F_* \cong R_*/I. We show that this action induces a free and transitive action of the group of quadratic forms QF(I/I^2[1]) on the set of equivalence classes of R-products on F. The characteristic bilinear form of F introduced by the authors in a previous paper is the natural obstruction to commutativity of F. We discuss the examples of the Morava K-theories K(n) and the 2-periodic Morava K-theories K_n. | Quadratic forms classify products on quotient ring spectra | 13,149 |
We study the moduli and determine a homotopy type of the space of all generalized Morse functions on d-manifolds for given d. This moduli space is closely connected to the moduli space of all Morse functions studied in the paper math.AT/0212321, and the classifying space of the corresponding cobordism category. | The moduli space of generalized Morse functions | 13,150 |
Variations on the notions of Reedy model structures and projective model structures on categories of diagrams in a model category are introduced. These allow one to choose only a subset of the entries when defining weak equivalences, or to use different model categories at different entries of the diagrams. As a result, a bisimplicial model category that can be used to recover the algebraic K-theory for any Waldhausen subcategory of a model category is produced. | On modified Reedy and modified projective model structures | 13,151 |
Let M be a connected, simply connected, closed and oriented manifold, and G a finite group acting on M by orientation preserving diffeomorphisms. In this paper we show an explicit ring isomorphism between the orbifold string topology of the orbifold [M/G] and the Hochschild cohomology of the dg-ring obtained by performing the smash product between the group G and the singular cochain complex of M. | Hochschild cohomology and string topology of global quotient orbifolds | 13,152 |
Let $[\rho_{i_k},[\rho_{i_{k-1}},...,[\rho_{i_{1}}, \rho_{i_2}] ...]]$ be an iterated commutator of self-maps $\rho_{i_j} : \Sigma {\Bbb H}P^\infty \to \Sigma {\Bbb H}P^\infty, j = 1,2, ..., k$ on the suspension of the infinite quaternionic projective space. In this paper, it is shown that the image of the homomorphism induced by the adjoint of this commutator is both primitive and decomposable. The main result in this paper asserts that the set of all homotopy types of spaces having the same $n$-type as the suspension of the infinite quaternionic projective space is the one element set consisting of a single homotopy type. Moreover, it is also shown that the group $\text{Aut}(\pi_{\leq n} (\Sigma {\Bbb H}P^\infty)/\text{torsion})$ of automorphisms is finite for $n \leq 9$, and infinite for $n \geq 13$, and that $\text{Aut}(\pi_{*} (\Sigma {\Bbb H}P^\infty)/\text{torsion})$ becomes non-abelian. | On the same $N$-type of the suspension of the infinite quaternionic
projective space | 13,153 |
We show that a map between fibrant objects in a closed model category is a weak equivalence if and only if it has the right homotopy extension lifting property with respect to all cofibrations. The dual statement holds for maps between cofibrant objects. | The HELP-Lemma and its converse in Quillen model categories | 13,154 |
On d\'emontre une conjecture due \'a N. Kuhn concernant la cohomologie singuli\'ere \'a coefficients mod p des espaces, comme module instable sur l'alg\'ebre de Steenrod. Notre d\'emonstration de ce r\'esultat, d\'ej\'a connu en caract\'eristique 2, fait appel \'a une m'ethode nouvelle, qui fonctionne en toute caracteristique. De cette mani\'ere on r\'etablit un r'esultat de [S98] dont la preuve est incompl\'ete dans le cas d'un nombre premier impair. ---- We settle a conjecture due to N. Kuhn about the mod p cohomology of spaces considered as unstable modules over the Steenrod algebra. This result is already known to hold in characteristic 2. The method presented here is essentially new and works for all characteristics. In doing so we fix a gap in [S98] concerning the odd prime case. | Applications depuis K(Z/p,2) et une conjecture de Kuhn | 13,155 |
We prove simplicial version of a classical theorem of Eilenberg in the equivariant context and give an alternative description of the simplicial version of Bredon-Illman cohomology with local coefficients, as introduced in [15], to derive a spectral sequence. | On a theorem of Eilenberg in simplicial Bredon-Illman cohomology with
local coefficients | 13,156 |
We recall how a description of local coefficients that Eilenberg introduced in the 1940s leads to spectral sequences for the computation of homology and cohomology with local coefficients. We then show how to construct new equivariant analogues of these spectral sequences and give a worked example of how to apply them in a computation involving the equivariant Serre spectral sequence. This paper contains some of the material in the author's Ph.D. thesis, which also discusses some results of L. Gaunce Lewis on the cohomology of complex projective spaces and corrects some flaws in his paper. | Equivariant Spectral Sequences for Local Coefficients | 13,157 |
On d\'emontre une conjecture due \`a N. Kuhn concernant la cohomologie singuli\`ere \`a coefficients mod p des espaces, comme module instable sur l'alg\`ebre de Steenrod. Notre d\'emonstration de ce r\'esultat, d\'ej\`a connu en caract\'eristique 2, fait appel \`a une m\'ethode nouvelle, qui fonctionne en toute caract\'eristique. De cette mani\`ere on r\'etablit le r\'esultat de [S98] dont la preuve est incompl\`ete dans le cas d'un nombre premier impair. We settle a conjecture due to N. Kuhn about the mod p cohomology of spaces considered as unstable modules over the Steenrod algebra. This result is already known to hold in characteristic 2. The method presented here is essentially new and works for all characteristics. In doing so we fix a gap in [S98] concerning the odd prime case. | Applications depuis K(Z/p, 2) et une conjecture de N. Kuhn | 13,158 |
The purpose of this paper is to investigate an algebraic version of the double complex transfer, in particular the classes in the two-line of the Adams-Novikov spectral sequence which are the image of comodule primitives of the MU-homology of the product of two copies of infinite complex projective space via the algebraic double transfer. These classes are analysed by two related approaches; the first, p-locally for an odd prime, by using the morphism induced in MU-homology by the chromatic factorization of the double transfer map together with the f'-invariant of Behrens (for p>=5). The second approach uses the algebraic double transfer and the f-invariant of Laures. | On the double transfer and the f-invariant | 13,159 |
A geometric approach to the stable homotopy groups of spheres is developed in this paper, based on the Pontryagin-Thom construction. The task of this approach is to obtain an alternative proof of the Hill-Hopkins-Ravenel theorem [H-H-R] on Kervaire invariants in all dimensions, except, possibly, a finite number of dimensions. In the framework of this approach, the Adams theorem on the Hopf invariant is studied, for all dimensions with the exception of 15, 31, 63, 127. The new approach is based on the methods of geometric topology. | Geometric approach to stable homotopy groups of spheres. I. The Hopf
invariant | 13,160 |
Every small category $C$ has a classifying space $BC$ associated in a natural way. This construction can be extended to other contexts and set up a fruitful interaction between categorical structures and homotopy types. In this paper we study the classifying space $B_2C$ of a 2-category $C$ and prove that, under certain conditions, the loop space $\Omega_c B_2C$ can be recovered up to homotopy from the endomorphisms of a given object. We also present several subsidiary results that we develop to prove our main theorem. | On the loop space of a 2-category | 13,161 |
We present an introduction to the manifold calculus of functors, due to Goodwillie and Weiss. Our perspective focuses on the role the derivatives of a functor F play in this theory, and the analogies with ordinary calculus. We survey the construction of polynomial functors, the classification of homogeneous functors, and results regarding convergence of the Taylor tower. We sprinkle examples throughout, and pay special attention to spaces of smooth embeddings. | Introduction to the manifold calculus of Goodwillie-Weiss | 13,162 |
We investigate the moduli sets of central extensions of H-spaces enjoying inversivity, power associativity and Moufang properties. By considering rational H-extensions, it turns out that there is no relationship between the first and the second properties in general. | On moduli subspaces of central extensions of rational H-spaces | 13,163 |
We study three connective versions of the spectrum for topological modular forms of level 3. All three were described briefly by Mahowald and Rezk in [Pure Appl Math Quar (2009)], but we add much detail to their discussion. Letting tmf(3) denote our connective model which is a ring spectrum, we compute the tmf(3)-homology of RP^infty. | Connective versions of TMF(3) | 13,164 |
We compare the structure of a mapping cone in the category Top^D of spaces under a space D with differentials in algebraic models like crossed complexes and quadratic complexes. Several subcategories of Top^D are identified with algebraic categories. As an application we show that there are exactly 16 essential self--maps of S^2 x S^2 fixing the diagonal. | Presentation of homotopy types under a space | 13,165 |
A quasitoric manifold is a $2n$-dimensional compact smooth manifold with a locally standard action of an $n$-dimensional torus whose orbit space is a simple polytope. In this article, we classify quasitoric manifolds with the second Betti number $\beta_2=2$ topologically. Interestingly, they are distinguished by their cohomology rings up to homeomorphism. | Topological classification of quasitoric manifolds with the second Betti
number 2 | 13,166 |
In this paper, we consider 2-dimensional precubical sets, which can be used to model systems of two concurrently executing processes. From the point of view of concurrency theory, two precubical sets can be considered equivalent if their geometric realizations have the same directed homotopy type relative to the extremal elements in the sense of P. Bubenik. We give easily verifiable conditions under which it is possible to reduce a 2-dimensional precubical set to an equivalent smaller one by collapsing an edge or eliminating a square and one or two free faces. We also look at some simple standard examples in order to illustrate how our results can be used to construct small models of 2-dimensional precubical sets. | Some collapsing operations for 2-dimensional precubical sets | 13,167 |
For twisted K-theory whose twist is classified by a degree three integral cohomology of infinite order, universal even degree characteristic classes are in one to one correspondence with invariant polynomials of Atiyah and Segal. The present paper describes the ring of these invariant polynomials by a basis and structure constants. | A basis of the Atiyah-Segal invariant polynomials | 13,168 |
We determine the algebra structure of the Hochschild cohomology of the singular cochain algebra with coefficients in a field on a space whose cohomology is a polynomial algebra. A spectral sequence calculation of the Hochschild cohomology is also described. In particular, when the underlying field is of characteristic two, we determine the associated bigraded Batalin-Vilkovisky algebra structure on the Hochschild cohomology of the singular cochain on a space whose cohomology is an exterior algebra. | The Hochschild cohomology ring of the singular cochain algebra of a
space | 13,169 |
Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete G-spectrum. If H and K are closed subgroups of G, with H normal in K, then, in general, the K/H-spectrum X^{hH} is not known to be a continuous K/H-spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (X^{hH})^{hK/H}. To address this situation, we define homotopy fixed points for delta-discrete G-spectra and show that the setting of delta-discrete G-spectra gives a good framework within which to work. In particular, we show that by using delta-discrete K/H-spectra, there is always an iterated homotopy fixed point spectrum, denoted (X^{hH})^{h_\delta K/H}, and it is just X^{hK}. Additionally, we show that for any delta-discrete G-spectrum Y, (Y^{h_\delta H})^{h_\delta K/H} \simeq Y^{h_\delta K}. Furthermore, if G is an arbitrary profinite group, there is a delta-discrete G-spectrum {X_\delta} that is equivalent to X and, though X^{hH} is not even known in general to have a K/H-action, there is always an equivalence ((X_\delta)^{h_\delta H})^{h_\delta K/H} \simeq (X_\delta)^{h_\delta K}. Therefore, delta-discrete L-spectra, by letting L equal H, K, and K/H, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete G-spectra. | Delta-discrete $G$-spectra and iterated homotopy fixed points | 13,170 |
Given a $C_\infty$ coalgebra $C_*$, a strict dg Hopf algebra $H_*$, and a twisting cochain $\tau:C_* \rightarrow H_*$ such that $Im(\tau) \subset Prim(H_*)$, we describe a procedure for obtaining an $A_\infty$ coalgebra on $C_* \otimes H_*$. This is an extension of Brown's work on twisted tensor products. We apply this procedure to obtain an $A_\infty$ coalgebra model of the chains on the free loop space $LM$ based on the $C_\infty$ coalgebra structure of $H_*(M)$ induced by the diagonal map $M \rightarrow M \times M$ and the Hopf algebra model of the based loop space given by $T(H_*(M)[-1])$. When $C_*$ has cyclic $C_\infty$ coalgebra structure, we describe an $A_\infty$ algebra on $C_* \otimes H_*$. This is used to give an explicit (non-minimal) $A_\infty$ algebra model of the string topology loop product. Finally, we discuss a representation of the loop product in principal $G$-bundles. | Homotopy Algebra Structures on Twisted Tensor Products and String
Topology Operations | 13,171 |
In this paper some axiomatic generalization (function of open subsets) of the relative Lyusternik-Schnirelmann category is considered, incorporating the sectional category and the Schwarz genus as well. For this function and a given continuous map of the underlying space to a finite-dimensional metric space some lower bounds on the value of this function on the (neighborhood of) preimage of some point are given. | The genus and the Lyusternik-Schnirelmann category of preimages | 13,172 |
Let E_n be the Lubin-Tate spectrum and let G_n be the nth extended Morava stabilizer group. Then there is a discrete G_n-spectrum F_n, with L_{K(n)}(F_n) \simeq E_n, that has the property that (F_n)^{hU} \simeq E_n^{hU}, for every open subgroup U of G_n. In particular, (F_n)^{hG_n} \simeq L_{K(n)}(S^0). More generally, for any closed subgroup H of G_n, there is a discrete H-spectrum Z_{n, H}, such that (Z_{n, H})^{hH} \simeq E_n^{hH}. These conclusions are obtained from results about consistent k-local profinite G-Galois extensions E of finite vcd, where L_k(-) is L_M(L_T(-)), with M a finite spectrum and T smashing. For example, we show that L_k(E^{hH}) \simeq E^{hH}, for every open subgroup H of G. | Obtaining intermediate rings of a local profinite Galois extension
without localization | 13,173 |
A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement $H_{i,\sigma}(\Lambda^{*,*}[P], d)$ over a principle ideal domain k and proved that $H_{*,*}(\Lambda[P], d)$ is isomorphic to the Tor of the corresponding face ring k(K) by the Taylor resolutions. As applications, we give methods to compute the ring structure of Tor_{*,*}^{k[x]}(k(K), k)$, $link_{K}\sigma$, $star_{K}\sigma$ and the cohomology of the generalized moment-angle complexes. | The homology of simplicial complement and the cohomology of the
moment-angle complexes | 13,174 |
Let $R$ be an $E_\infty$-ring spectrum. Given a map $\zeta$ from a space $X$ to $BGL_1R$, one can construct a Thom spectrum, $X^\zeta$, which generalises the classical notion of Thom spectrum for spherical fibrations in the case $R=S^0$, the sphere spectrum. If $X$ is a loop space ($\simeq \Omega Y$) and $\zeta$ is homotopy equivalent to $\Omega f$ for a map $f$ from $Y$ to $B^2GL_1R$, then the Thom spectrum has an $A_\infty$-ring structure. The Topological Hochschild Homology of these $A_\infty$-ring spectra is equivalent to the Thom spectrum of a map out of the free loop space of $Y$. This paper considers the case $X=S^1$, $R=K_p^\wedge$, the p-adic $K$-theory spectrum, and $\zeta = 1-p \in \pi_1BGL_1K_p^\wedge$. The associated Thom spectrum $(S^1)^\zeta$ is equivalent to the mod p $K$-theory spectrum $K/p$. The map $\zeta$ is homotopy equivalent to a loop map, so the Thom spectrum has an $A_\infty$-ring structure. I will compute $\pi_*THH^{K_p^\wedge}(K/p)$ using its description as a Thom spectrum. | Topological Hochschild Homology of $K/p$ as a $K_p^\wedge$ module | 13,175 |
The intention of this article is to make an attempt of classification of transitive Lie algebroids and on this basis to construct a classifying space. The realization of the intention allows to describe characteristic classes of transitive Lie algebroids form the point of view a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles. | Transitive Lie algebroids - categorical point of view | 13,176 |
We show that every rank two $p$-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group $G$ on a manifold $M$, we construct a smooth free action on $M \times \bbS ^{n_1} \times \dots \times \bbS ^{n_k}$ when the set of isotropy subgroups of the $G$-action on $M$ can be associated to a fusion system satisfying certain properties. Another consequence of this construction is that if $G$ is an (almost) extra-special $p$-group of rank $r$, then it acts freely and smoothly on a product of $r$ spheres. | Fusion systems and constructing free actions on products of spheres | 13,177 |
Two constructions of a Lie model of the interval were performed by R. Lawrence and D. Sullivan. The first model uses an inductive process and the second one comes directly from solving a differential equation. They conjectured that these two models are the same. We prove this conjecture here. | Lawrence-Sullivan models for the interval | 13,178 |
Let $X$ be a complete $\Q$-factorial toric variety of dimension $n$ and $\del$ the fan in a lattice $N$ associated to $X$. For each cone $\sigma$ of $\del$ there corresponds an orbit closure $V(\sigma)$ of the action of complex torus on $X$. The homology classes $\{[V(\sigma)]\mid \dim \sigma=k\}$ form a set of specified generators of $H_{n-k}(X,\Q)$. It is shown that, given $\alpha\in H_{n-k}(X,\Q)$, there is a canonical way to express $\alpha$ as a linear combination of the $[V(\sigma)]$ with coefficients in the field of rational functions of degree $0$ on the Grassmann manifold of $(n-k+1)$-planes in $N_\Q$. This generalizes Morelli's formula for $\alpha$ the $(n-k)$-th component of the Todd homology class of the variety $X$. | On a Morelli type expression of cohomology classes of toric varieties | 13,179 |
A central extension of the form $E: 0 \to V \to G \to W \to 0$, where $V$ and $W$ are elementary abelian 2-groups, is called Bockstein closed if the components $q_i \in H^*(W, \FF_2)$ of the extension class of $E$ generate an ideal which is closed under the Bockstein operator. In this paper, we study the cohomology ring of $G$ when $E$ is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring of $G$ has a simple form and it is easy to calculate. The main result of the paper is the calculation of the Bocksteins of the generators of the mod-2 cohomology ring using an Eilenberg-Moore spectral sequence. We also find an interpretation of the second page of the Bockstein spectral sequence in terms of a new cohomology theory that we define for Bockstein closed quadratic maps $Q : W \to V$ associated to the extensions $E$ of the above form. | Bockstein Closed 2-Group Extensions and Cohomology of Quadratic Maps | 13,180 |
The stable systolic category of a closed manifold M indicates the complexity in the sense of volume. This is a homotopy invariant, even though it is defined by some relations between homological volumes on M. We show an equality of the stable systolic category and the real cup-length for the product of arbitrary finite dimensional real homology spheres. Also we prove the invariance of the stable systolic category under the rational equivalences for orientable 0-universal manifolds. | Stable systolic category of the product of spheres | 13,181 |
We give an independent, and perhaps somewhat simplified, description of the product in negative Tate-cohomology (the generalised version for compact Lie-groups). We describe, but do not compute, the corresponding action of the Dyer--Lashof-algebra, using the linear-isometries operad. | The linear isometries operad in Lie--Tate homology | 13,182 |
Let M be a homogeneous space admitting a left translation by a connected Lie group G. The adjoint to the action gives rise to a map from G to the monoid of self-homotopy equivalences of M.The purpose of this paper is to investigate the injectivity of the homomorphism which is induced by the adjoint map on the rational homotopy. In particular, the visible degrees are determined explicitly for all the cases of simple Lie groups and their associated homogeneous spaces of rank one which are classified by Oniscik. | Rational visibility of a Lie group in the monoid of self-homotopy
equivalences of a homogeneous space | 13,183 |
We compute the monoid of essential self-maps of of the product of two n-spheres fixing the diagonal. More generally, we consider products S x S, where S is a suspension. Essential self-maps of S x S demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions. | Self-Maps of the Product of Two Spheres Fixing the Diagonal | 13,184 |
We give a necessary and sufficient condition for the orientability of a locally standard 2-torus manifold with a fixed point which generalizes previous results of Nakayama-Nishimura in 2005 and Soprunova-Sottile in 2013. We construct manifolds with boundary where the boundary is a disjoint union of locally standard 2-torus manifolds. We discuss equivariant oriented cobordism class of locally standard 2-torus manifolds. | Orientability and equivariant oriented cobordism of 2-torus manifolds | 13,185 |
We describe an iterable construction of THH for an E_n ring spectrum. The reduced version is an iterable bar construction and its n-th iterate gives a model for the shifted cotangent complex at the augmentation, representing reduced topological Quillen homology of an augmented E_n algebra. | Homology of E_n Ring Spectra and Iterated THH | 13,186 |
Let $p$ be a prime. We calculate the connective unitary K-theory of the smash product of two copies of the classifying space for the cyclic group of order $p$, using a K\"{u}nneth formula short exact sequence. As a corollary, using the Bott exact sequence and the mod $2$ Hurewicz homomorphism we calculate the connective orthogonal K-theory of the smash product of two copies of the classifying space for the cyclic group of order two. | Ossa's Theorem via the Kunneth formula | 13,187 |
We show that three different kinds of cohomology - Baues-Wirsching cohomology, the (S,O)-cohomology of Dwyer-Kan, and the Andre-Quillen cohomology of a Pi-algebra - are isomorphic, under certain assumptions. This is then used to identify the cohomological obstructions in three general approaches to realizability problems: the track category version of Baues-Wirsching, the diagram rectifications of Dwyer-Kan-Smith, and the Pi-Algebra realization of Dwyer-Kan-Stover. Our main tool in this identification is the notion of a mapping algebra: a simplicially enriched version of an algebra over a theory. | Comparing cohomology obstructions | 13,188 |
We introduce the category Pstem[n] of n-stems, with a functor P[n] from spaces to Pstem[n]. This can be thought of as the n-th order homotopy groups of a space. We show how to associate to each simplicial n-stem Q an (n+1)-truncated spectral sequence. Moreover, if Q=P[n]X is the Postnikov n-stem of a simplicial space X, the truncated spectral sequence for Q is the truncation of the usual homotopy spectral sequence of X. Similar results are also proven for cosimplicial n-stems. They are helpful for computations, since n-stems in low degrees have good algebraic models. | Stems and Spectral Sequences | 13,189 |
Let G be an exceptional Lie group with a maximal torus T. Based on Schubert calculus on the flag manifold G/T we have described the integral cohomology ring H*(G) by explicitely constructed generators in [DZ2], and determined the structure of H*(G;F_{p}) as a Hopf algebra over the Steenrod algebra in[DZ3]. In this sequel to [DZ2], [DZ3] we obtain the near--Hopf ring structure on H*(G). | The near-Hopf ring structure on the integral cohomology of 1-connected
Lie groups | 13,190 |
We classify the most common local forms of smooth maps from a smooth manifold L to the plane. The word "local" can refer to locations in the source L, but also to locations in the target. The first point of view leads us to a classification of certain germs of maps, which we review here although it is very well known. The second point of view leads us to a classification of certain multigerms of maps. | Smooth maps to the plane and Pontryagin classes, Part I: Local aspects | 13,191 |
We introduce a large scale analogue of the classical fixed-point property for continuous maps, which shall apply to coarse maps. We also develop a coarse version of degree for coarse maps on Euclidean spaces. Then, applying a coarse degree-theoretic argument, we prove that every coarse map from a Euclidean half-space to itself has the coarse fixed-point property. | A Degree-Theoretic Proof of a Coarse Fixed Point Principle | 13,192 |
We show that the well-known fact that the equivariant cohomology of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with maximal isotropy rank. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces. | Torsion in equivariant cohomology and Cohen-Macaulay G-actions | 13,193 |
We show the $\TT^2$-cobordism group of the category of 4-dimensional quasitoric manifolds is generated by the $\TT^2$-cobordism classes of $\CP^2$. We construct nice oriented $\TT^2$ manifolds with boundary where the boundary is the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds. | $\TT^2$-cobordism of Quasitoric 4-Manifolds | 13,194 |
On the tensor product of two homotopy Gerstenhaber algebras we construct a Hirsch algebra structure which extends the canonical dg algebra structure. Our result applies more generally to tensor products of "level 3 Hirsch algebras" and also to the Mayer-Vietoris double complex. | Tensor products of homotopy Gerstenhaber algebras | 13,195 |
We study the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime 2. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. We relate the Goodwillie filtration to the P map, and the Goodwillie differentials to the H map. Furthermore, we study an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. We show that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. We use our theory to re-compute the 2-primary unstable stems through the Toda range (up to the 19-stem). We also study the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching's operad structure on the derivatives of the identity. These operations act on the mod 2 stable homology of the Goodwillie layers of any functor from spaces to spaces. | The Goodwillie tower and the EHP sequence | 13,196 |
We develop the properties of the $n$-th sequential topological complexity $TC_n$, a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in robotics. We exhibit close connections of $TC_n(X)$ to the Lusternik-Schnirelmann category of cartesian powers of $X$, to the cup-length of the diagonal embedding $X\hookrightarrow X^n$, and to the ratio between homotopy dimension and connectivity of $X$. We fully compute the numerical value of $TC_n$ for products of spheres, closed 1-connected symplectic manifolds, and quaternionic projective spaces. Our study includes two symmetrized versions of $TC_n(X)$. The first one, unlike Farber-Grant's symmetric topological complexity, turns out to be a homotopy invariant of $X$; the second one is closely tied to the homotopical properties of the configuration space of cardinality-$n$ subsets of $X$. Special attention is given to the case of spheres. | Higher topological complexity and its symmetrization | 13,197 |
In this paper, we discuss the construction of classifying spaces of fibre sequences in model categories of simplicial sheaves. One construction proceeds via Brown representability and provides a classification in the pointed model category. The second construction is given by the classifying space of the monoid of homotopy self-equivalences of a simplicial sheaf and provides the unpointed classification. | Classifying spaces and fibrations of simplicial sheaves | 13,198 |
We define model structures on exact categories which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly idempotent complete we get Hovey's one-to-one correspondence between model structures and complete cotorsion pairs. We classify the right and left homotopy relation in terms of the cotorsion pairs and look at examples of exact model structures. In particular, we see that given any hereditary abelian model category, the full subcategories of cofibrant, fibrant and cofibrant-fibrant subobjects each have natural exact model structures equivalent to the original model structure. These model structures each have interesting characteristics. For example, the cofibrant-fibrant subobjects form a Frobenius category whose stable category is the same thing as the homotopy category of its model structure. | Model Structures on Exact Categories | 13,199 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.