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We show that the sum over planar trees formula of Kontsevich and Soibelman transfers C-infinity structures along a contraction. Applying this result to a cosimplicial commutative algebra A^* over a field of characteristic zero, we exhibit a canonical unital C-infinity structure on Tot(A^*), which is unital if A^* is; in particular, we obtain a canonical C-infinity structure on the cochain complex of a simplicial set. | Transferring homotopy commutative algebraic structures | 12,800 |
The formula is $\partial{e}=({\rm ad}_e)b+\sum_{i=0}^\infty{\frac{B_i}{i!}}({\rm ad}_e)^i(b-a)\>,$ with $\partial{a}+{1\over2}[a,a] =0$ and $\partial{b}+{1\over2}[b,b] =0$, where $a$, $b$ and $e$ in degrees $-1$, $-1$ and 0 are the free generators of a completed free graded Lie algebra $L[a,b,e]$. The coefficients are defined by ${x\over{e^x-1}}=\sum_{n=0}^\infty{B_n\over{}n!}x^n$. The theorem is that (I) this formula for $\partial$ on generators extends to a derivation of square zero on $L[a,b,e]$, (II) the formula for $\partial{e}$ is unique satisfying the first property, once given the formulae for $\partial{a}$ and $\partial{b}$, along with the condition that the "flow" generated by $e$ moves $a$ to $b$ in unit time. The immediate significance of this formula is that it computes the infinity cocommutative coalgebra structure on the chains of the closed interval. It may be derived and proved using the geometrical idea of flat connections and one parameter groups or flows of gauge transformations. The deeper significance of such general DGLAs which want to combine deformation theory and rational homotopy theory is proposed as a research problem. | A formula for topology/deformations and its significance | 12,801 |
Let M be a Poincare duality space of dimension at least four. In this paper we describe a complete obstruction to realizing the diagonal map M -> M x M by a Poincare embedding. The obstruction group depends only on the fundamental group and the parity of the dimension of M. | Poincare Complex Diagonals | 12,802 |
We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study classical invariants of general topological spaces. | Minimal Finite Models | 12,803 |
Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension 1\to Z\to \hat{K}\to K\to 1 of K. It is a classical question whether there exists a \hat{K}-principal bundle \hat{P} on M such that \hat{P}/Z is isomorphic to P. Neeb defines in this context a crossed module of topological Lie algebras whose cohomology class [\omega_{\rm top alg}] is an obstruction to the existence of \hat{P}. In the present paper, we show that [\omega_{\rm top alg}] is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes. | Obstruction classes of crossed modules of Lie algebroids and Lie
groupoids linked to existence of principal bundles | 12,804 |
In this work, we study topological properties of surface bundles, with an emphasis on surface bundles with a spin structure. We develop a criterion to decide whether a given manifold bundle has a spin structure and specialize it to surface bundles. We study examples of surface bundles, in particular sphere and torus bundles and surface bundles induced by actions of finite groups on Riemann surfaces. The examples are used to show that the obstruction cohomology class against the existence of a spin structure is nonzero. We develop a connection between the Atiyah-Singer index theorem for families of elliptic operators and the modern homotopy theory of moduli spaces of Riemann surfaces due to Tillmann, Madsen and Weiss. This theory and the index theorem is applied to prove that the tautological classes of spin surface bundles satisfy certain divisibility relations. The result is that the divisibility improves, compared with the non-spin-case, by a certain power of 2. The explicit computations for sphere bundles are used in the proof of the divisibility result. In the last chapter, we use actions of certain finite groups to construct explicit torsion elements in the homotopy groups of the mapping class and compute their order, which relies on methods from algebraic $K$-theory and on the Madsen-Weiss theorem. | Characteristic classes of spin surface bundles. Appliations of the
Madsen-Weiss theory | 12,805 |
Let us take for granted that L_{K(n)}S^0 --> E_n is some kind of a G_n-Galois extension. Of course, this is in the setting of continuous G_n-spectra. How much structure does this continuous G-Galois extension have? How much structure does one want to build into this notion to obtain useful conclusions? If the author's conjecture that ``E_n/I, for a cofinal collection of I's, is a discrete G_n-symmetric ring spectrum" is true, what additional structure does this give the continuous G_n-Galois extension? Is it useful or merely beautiful? This paper is an exploration of how to answer these questions. This preprint arose as a letter to John Rognes, whom he thanks for a helpful conversation in Rosendal. This paper was written before John's preprints (the initial version and the final one) on Galois extensions were available. | Rognes's theory of Galois extensions and the continuous action of G_n on
E_n | 12,806 |
The aim of this paper is to present a very simple original, purely formal, proof of Quillen's adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an abstract adjunction theorem for "absolute" derived functors. In contrast with all known proofs, the explicit construction of the derived functors is not used. | Quillen's adjunction theorem for derived functors, revisited | 12,807 |
Let A be an A_\infty ring spectrum. We use the description from [2] of the cyclic bar and cobar construction to give a direct definition of topological Hochschild homology and cohomology of A using the Stasheff associahedra and another family of polyhedra called cyclohedra. This construction builds the maps making up the A_\infty structure into THH(A), and allows us to study how THH(A) varies over the moduli space of A_\infty structures on A. As an example, we study how topological Hochschild cohomology of Morava K-theory varies over the moduli space of A_\infty structures and show that in the generic case, when a certain matrix describing the noncommutativity of the multiplication is invertible, topological Hochschild cohomology of 2-periodic Morava K-theory is the corresponding Morava E-theory. If the A_\infty structure is ``more commutative'', topological Hochschild cohomology of Morava K-theory is some extension of Morava E-theory. | Topological Hochschild homology and cohomology of A_\infty ring spectra | 12,808 |
We develop the theory of CW(A)-complexes, which generalizes the classical theory of CW-complexes, keeping the geometric intuition of J.H.C. Whitehead's original theory. We obtain this way generalizations of classical results, such as Whitehead Theorem, which allow a deeper insight in the homotopy properties of these spaces. | A geometric decomposition of spaces into cells of different types | 12,809 |
We say that a space X admits a homology exponent if there exists an exponent for the torsion subgroup of the integral homology. Our main result states if an H-space of finite type admits a homology exponent, then either it is, up to 2-completion, a product of spaces of the form BZ/2^r, S^1, K(Z, 2), and K(Z,3), or it has infinitely many non-trivial homotopy groups and k-invariants. We then show with the same methods that simply connected $H$-spaces whose mod 2 cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod 2 finite H-spaces with copies of K(Z, 2) and K(Z,3). | Homology exponents for H-spaces | 12,810 |
We study a collection of operations on the cohomotopy of any space, with which it becomes a "beta-ring", an algebraic structure analogous to a lambda-ring. In particular, this ring possesses Adams operations, represented by maps on the infinite loop space of the sphere spectrum. We compute their effect in homotopy on the image of J, and in mod 2 cohomology. The motivation comes from the interpretation of the symmetric group as the general linear group of the "field with one element", which leads to an analogy between cohomotopy and algebraic K-theory. A good deal of this article may be considered as a survey of the theory of beta-rings. | Adams operations in cohomotopy | 12,811 |
The goal of this memoir is to prove that the bar complex B(A) of an E-infinity algebra A is equipped with the structure of a Hopf E-infinity algebra, functorially in A. We observe in addition that such a structure is homotopically unique provided that we consider unital operads which come equipped with a distinguished 0-ary operation that represents the natural unit of the bar complex. Our constructions rely on a Reedy model category for unital Hopf operads. For our purpose we define a unital Hopf endomorphism operad which operates functorially on the bar complex and which is universal with this property. Then we deduce our structure results from operadic lifting properties. To conclude this memoir we hint how to make our constructions effective and explicit. | The universal Hopf operads of the bar construction | 12,812 |
We prove that every commutative differential graded algebra whose cohomology is a simply-connected Poincare duality algebra is quasi-isomorphic to one whose underlying algebra is simply-connected and satisfies Poincare duality in the same dimension. This has application in particular to the study of CDGA models of configuration spaces on a closed manifold. | Poincare duality and commutative differential graded algebras | 12,813 |
We show that the Bousfield-Kan spectral sequence which computes the rational homotopy groups of the space of long knots in ${\mathbb R}^d$, where $d\ge 4$, collapses at the $E^2$ page. The main ingredients in the proof are Sinha's cosimplicial model for the space of long knots and a coformality result for the little balls operad. | Coformality and the rational homotopy groups of spaces of long knots | 12,814 |
Let $f,g: X\to G/K$ be maps from a closed connected orientable manifold $X$ to an orientable coset space $M=G/K$ where $G$ is a compact connected Lie group, $K$ a closed subgroup and $\dim X=\dim M$. In this paper, we show that if $L(f,g)=0$ then $N(f,g)=0$; if $L(f,g)\ne 0$ then $N(f,g)=R(f,g)$ where $L(f,g), N(f,g)$, and $R(f,g)$ denote the Lefschetz, Nielsen, and Reidemeister coincidence numbers of $f$ and $g$, respectively. When $\dim X> \dim M$, we give conditions under which $N(f,g)=0$ implies $f$ and $g$ are deformable to be coincidence free. | Jiang-type theorems for coincidences of maps into homogeneous spaces | 12,815 |
Given a topological group G, its orbit category Orb_G has the transitive G-spaces G/H as objects and the G-equivariant maps between them as morphisms. A well known theorem of Elmendorf then states that the category of G-spaces and the category of contravariant functors Func(Orb_G,Spaces) have equivalent homotopy theories. We extend this result to the context of orbispaces, with the role of Orb_G now played by a category whose objects are topological groups and whose morphisms are given by Hom(H,G) = Mono(H,G) x_G EG. On our way, we endow the category of topological groupoids with notions of weak equivalence, fibrant objects, and cofibrant objects, and show that it then shares many of the properties of a Quillen model category. | Homotopy Theory of Orbispaces | 12,816 |
The complex affine quadric $Q^{m}=\{z\in {\Bbb C}^{m+1}\mid z_{1}^{2}+...+z_{m+1}^{2}=1\}$ deforms by retraction onto $S^{m}$; this allows us to identify $[Q^{k},Q^{n}]$ and $[S^{k},S^{n}]=\pi_{k}(S^{n})$. Thus one will say that an element of $\pi_{k}(S^{n})$ is complex representable if there exists a complex polynomial map from $Q^{k}$ to $Q^{n}$ corresponding to this class. In this Note we show that $\pi_{n+1}(S^{n})$ and $\pi_{n+2}(S^{n})$ are complex representable. | Complex Polynomial Representation of $π_{n+1}(s^{n})$ and
$π_{n+2}(s^{n})$ | 12,817 |
The virtual cohomology of an orbifold is a ring structure on the cohomology of the inertia orbifold whose product is defined via the pull-push formalism and the Euler class of the excess intersection bundle. In this paper we calculate the virtual cohomology of a large family of orbifolds, including the symmetric product. | Orbifold Virtual Cohomology of the Symmetric Product | 12,818 |
We study the minimal dimension of the classifying space of the family of virtually cyclic subgroups of a discrete group. We give a complete answer for instance if the group is virtually poly-Z, word-hyperbolic or countable locally virtually cyclic. We give examples of groups for which the difference of the minimal dimensions of the classifying spaces of virtually cyclic and of finite subgroups is -1, 0 and 1, and show in many cases that no other values can occur. | On the classifying space of the family of virtually cyclic subgroups | 12,819 |
A global action is the algebraic analogue of a topological manifold. This construction was introduced in first place by A. Bak as a combinatorial approach to K-Theory and the concept was later generalized by Bak, Brown, Minian and Porter to the notion of groupoid atlas. In this paper we define and investigate homotopy invariants of global actions and groupoid atlases, such as the strong fundamental groupoid, the weak and strong nerves, classifying spaces and homology groups. We relate all these new invariants to classical constructions in topological spaces, simplicial complexes and simplicial sets. This way we obtain new combinatorial formulations of classical and non classical results in terms of groupoid atlases. | Classical invariants for global actions and groupoid atlases | 12,820 |
A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper. | One-Dimensional Reduction of Multidimensional Persistent Homology | 12,821 |
We apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type U(1,n-1). These cohomology theories of topological automorphic forms (TAF) are related to Shimura varieties in the same way that TMF is related to the moduli space of elliptic curves. We study the cohomology operations on these theories, and relate them to certain Hecke algebras. We compute the K(n)-local homotopy types of these cohomology theories, and determine that K(n)-locally these spectra are given by finite products of homotopy fixed point spectra of the Morava E-theory E_n by finite subgroups of the Morava stabilizer group. We construct spectra Q_U(K) for compact open subgroups K of certain adele groups, generalizing the spectra Q(l) studied by the first author in the modular case. We show that the spectra Q_U(K) admit finite resolutions by the spectra TAF, arising from the theory of buildings. We prove that the K(n)-localizations of the spectra Q_U(K) are finite products of homotopy fixed point spectra of E_n with respect to certain arithmetic subgroups of the Morava stabilizer groups, which N. Naumann has shown (in certain cases) to be dense. Thus the spectra Q_U(K) approximate the K(n)-local sphere to the same degree that the spectra Q(l) approximate the K(2)-local sphere. | Topological automorphic forms | 12,822 |
We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it. The conjecture says that a map from a finite CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial. As a corollary we show that in the above situation all components of non-zero degree maps in the space of maps from X to Y are contractible. | On a conjecture of Daniel H. Gottlieb | 12,823 |
The structure of a $k$-fold monoidal category as introduced by Balteanu, Fiedorowicz, Schw\"anzl and Vogt can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of ($n$-fold) operads in a $k$-fold monoidal category which generalizes the definition of operads in a braided category. Furthermore, the inheritance of structure by the category of operads is actually an inheritance of iterated monoidal structure, decremented by at least two iterations. We prove that the category of $n$-fold operads in a $k$-fold monoidal category is itself a $(k-n)$-fold monoidal, strict 2-category, and show that $n$-fold operads are automatically $(n-1)$-fold operads. We also introduce a family of simple examples of $k$-fold monoidal categories and classify operads in the example categories. | Operads in iterated monoidal categories | 12,824 |
In 1999 Chas and Sullivan discovered that the homology H_*(LX) of the space of free loops on a closed oriented smooth manifold X has a rich algebraic structure called string topology. They proved that H_*(LX) is naturally a Batalin-Vilkovisky (BV) algebra. There are several conjectures connecting the string topology BV algebra with algebraic structures on the Hochschild cohomology of algebras related to the manifold X, but none of them has been verified for manifolds of dimension n>1. In this work we study string topology in the case when X is aspherical (i.e. its homotopy groups \pi_i(X) vanish for i > 1). In this case the Hochschild cohomology Gerstenhaber algebra HH^*(A) of the group algebra A of the fundamental group of X has a BV structure. Our main result is a theorem establishing a natural isomorphism between the Hochschild cohomology BV algebra HH^*(A) and the string topology BV algebra H_*(LX). In particular, for a closed oriented surface X of hyperbolic type we obtain a complete description of the BV algebra operations on H_*(LX) and HH^*(A) in terms of the Goldman bracket of loops on X. The only manifolds for which the BV algebra structure on H_*(LX) was known before were spheres and complex Stiefel manifolds. Our proof is based on a combination of topological and algebraic constructions allowing us to compute and compare multiplications and BV operators on both H_*(LX) and HH^*(A). | The string topology BV algebra, Hochschild cohomology and the Goldman
bracket on surfaces | 12,825 |
In this paper we consider a class of connected closed $G$-manifolds with a non-empty finite fixed point set, each $M$ of which is totally non-homologous to zero in $M_G$ (or $G$-equivariantly formal), where $G={\Bbb Z}_2$. With the help of the equivariant index, we give an explicit description of the equivariant cohomology of such a $G$-manifold in terms of algebra, so that we can obtain analytic descriptions of ring isomorphisms among equivariant cohomology rings of such $G$-manifolds, and a necessary and sufficient condition that the equivariant cohomology rings of such two $G$-manifolds are isomorphic. This also leads us to analyze how many there are equivariant cohomology rings up to isomorphism for such $G$-manifolds in 2- and 3-dimensional cases. | Equivariant cohomology and analytic descriptions of ring isomorphisms | 12,826 |
We determine the Batalin-Vilkovisky Lie algebra structure for the integral loop homology of special unitary groups and complex Stiefel manifolds. It is shown to coincide with the Poisson algebra structure associated to a certain odd symplectic form on a super vector space for which loop homology is the super algebra of functions. Over rationals, the loop homology of the above spaces splits into a tensor product of simple BV algebras, and it is shown to contain a super Lie algebra. | Batalin-Vilkovisky Lie Algebra Structure on the Loop Homology of Complex
Stiefel Manifolds | 12,827 |
By studying the group of self homotopy equivalences of the localization (at a prime $p$ and/or zero) of some aspherical complexes, we show that, contrary to the case when the considered space is a nilpotent complex, $\mathcal{E}_{\#}^m (X_{p})$ is in general different from $\mathcal{E}_{\#}^m (X)_{p}$. That is the case even when $X=K(G,1)$ is a finite complex and/or $G$ satisfies extra finiteness or nilpotency conditions, for instance, when $G$ is finite or virtually nilpotent. | Homotopy equivalences of localized aspherical complexes | 12,828 |
We generalize Berger and Moerdijk's results on axiomatic homotopy theory for operads to the setting of enriched symmetric monoidal model categories, and show how this theory applies to orthogonal spectra. In particular, we provide a symmetric fibrant replacement functor for the positive stable model structure. | Model structure on operads in orthogonal spectra | 12,829 |
In this paper we study the nilpotency of certain groups of self homotopy equivalences. Our main goal is to extend, to localized homotopy groups and/or homotopy groups with coefficients, the general principle of Dror and Zabrodsky by which a group of self homotopy equivalences of a finite space which acts nilpotently on the homotopy groups is itself nilpotent. | Nilpotency of self homotopy equivalences with coefficients | 12,830 |
Cofibrations are defined in the category of Fr\"olicher spaces by weakening the analog of the classical definition to enable smooth homotopy extensions to be more easily constructed, using flattened unit intervals. We later relate smooth cofibrations to smooth neighborhood deformation retracts. The notion of smooth neighborhood deformation retract gives rise to an analogous result that a closed Fr\"olicher subspace $A$ of the Fr\"olicher space $X$ is a smooth neighborhood deformation retract of $X$ if and only if the inclusion $i: A\hookrightarrow X$ comes from a certain subclass of cofibrations. As an application we construct the right Puppe sequence. | Cofibrations in the Category of Frolicher Spaces. Part I | 12,831 |
In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RP^n with n as small as 48, and yet it was stated there that the first new result occurred when n=1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f^*(X)=X_1+X_2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in the original paper. Finally, we present a complete determination of tmf^{8*}(RP^\infty\times RP^\infty) and tmf^*(CP^\infty\times CP^\infty) in positive dimensions. | Nonimmersions of RP^n implied by tmf, revisited | 12,832 |
For an algebraically closed field K with ch K \not = 2, we determine the Chow ring of the moduli space of holomorphic bundles on a projective plane with the structure group SO(n,K) and half the first Pontryagin index being equal to 1, each of which is trivial on a fixed line and has a fixed holomorphic trivialization there. | The Chow ring of the moduli space and its related homogeneous space of
bundles on P^2 with charge 1 | 12,833 |
We extend Hendriks' classification theorem and Turaev's realisation and splitting theorems for Poincare duality complexes in dimension three to the relative case of Poincare duality pairs. The results for Poincare duality complexes are recovered by restricting the results to the case of Poincare duality pairs with empty boundary. Up to oriented homotopy equivalence, three-dimensional Poincare duality pairs are classified by their fundamental triple consisting of the fundamental group system, the orientation character and the image of the fundamental class under the classifying map. Using the derived module category we provide necessary and sufficient conditions for a given triple to be realised by a three-dimensional Poincare duality pair. The results on classification and realisation yield splitting or decomposition theorems for three-dimensional Poincare duality pairs, that is, conditions under which a given three-dimensional Poincare duality pair decomposes as interior or boundary connected sum of two three-dimensional Poincare duality pairs. | Poincare Duality Pairs in Dimension Three | 12,834 |
In the theory of operads we consider functors of generalized symmetric powers defined by sums of coinvariant modules under actions of symmetric groups. One observes classically that the construction of symmetric functors provides an isomorphism from the category of symmetric modules to a subcategory of the category of functors on the base category. The purpose of this book is to obtain a similar relationship for functors on a category of algebras over an operad. We observe that right modules over operads, symmetric modules equipped with a right operad action, give rise to functors on categories of algebras and we prove that this construction yields an embedding of categories. Then we check that right modules over operads form a model category. In addition we prove that weak-equivalences of right modules correspond to pointwise weak-equivalences at the functor level. As a conclusion, we obtain that right modules over operads supply good models for the homotopy of associated functors on algebras over operads. | Modules over operads and functors | 12,835 |
Let $X$ be a nilpotent space such that there exists $p\geq 1$ with $H^p(X,\mathbb Q) \ne 0$ and $H^n(X,\mathbb Q)=0$ if $n>p$. Let $Y$ be a m-connected space with $m\geq p+1$ and $H^*(Y,\mathbb Q)$ is finitely generated as algebra. We assume that $X$ is formal and there exists $p$ odd such that $H^p(X,\mathbb Q) \ne 0$. We prove that if the space $\mathcal F(X,Y)$ of continuous maps from $X$ to $Y$ is formal, then $Y$ has the rational homotopy type of a product of Eilenberg Mac Lane spaces. At the opposite, we exhibit an example of a formal space $\mathcal F(S^2,Y)$ where $Y$ is not rationally equivalent to a product of Eilenberg Mac Lane spaces. | Formality of function spaces | 12,836 |
Let A be a subspace arrangement with a geometric lattice such that codim(x) > 1 for every x in A. Using rational homotopy theory, we prove that the complement M(A) is rationally elliptic if and only if the sum of the orthogonal subspaces is a direct sum. The homotopy type of M(A) is also given: it is a product of odd dimensional spheres. Finally, some other equivalent conditions are given, such as Poincare duality. Those results give a complete description of arrangements (with geometric lattice and with the codimension condition on the subspaces) such that M(A) is rationally elliptic, and show that most arrangements have an hyperbolic complement. | Rational homotopy type of subspace arrangements with a geometric lattice | 12,837 |
We construct an abelian category A(G) of sheaves over a category of closed subgroups of the r-torus G and show it is of finite injective dimension. It can be used as a model for rational $G$-spectra in the sense that there is a homology theory \piA_*: G-spectra/Q --> A(G) on rational G-spectra with values in A(G), and the associated Adams spectral sequence converges for all rational $G$-spectra and collapses at a finite stage. | Rational torus-equivariant homotopy I: calculating groups of stable maps | 12,838 |
The circle-equivariant spectrum MString_C is the equivariant analogue of the cobordism spectrum MU<6> of stably almost complex manifolds with c_1=c_2=0. Given a rational elliptic curve C, the second author has defined a ring T-spectrum EC representing the associated T-equivariant elliptic cohomology. The core of the present paper is the construction, when C is a complex elliptic curve, of a map of ring T-spectra MString_C --> EC which is the rational equivariant analogue of the sigma orientation of Ando-Hopkins-Strickland. We support this by a theory of characteristic classes for calculation, and a conceptual description in terms of algebraic geometry. In particular, we prove a conjecture of the first author. | Circle-equivariant classifying spaces and the rational equivariant sigma
genus | 12,839 |
The result of this paper is the determination of the cohomology of Artin groups of type A_n, B_n and \tilde{A}_{n} with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B_n with coefficients over the module \Q[q^{\pm 1},t^{\pm 1}]. Here the first (n-1) standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type \tilde{A}_{n} as well as the cohomology of the classical braid group {Br}_{n} with coefficients in the n-dimensional representation presented in \cite{tong}. The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(\pi,1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived. | Cohomology of affine Artin groups and applications | 12,840 |
In this paper we prove that the complement to the affine complex arrangement of type \widetilde{B}_n is a K(\pi, 1) space. We also compute the cohomology of the affine Artin group G of type \widetilde{B}_n with coefficients over several interesting local systems. In particular, we consider the module Q[q^{\pm 1}, t^{\pm 1}], where the first n-standard generators of G act by (-q)-multiplication while the last generator acts by (-t)-multiplication. Such representation generalizes the analog 1-parameter representation related to the bundle structure over the complement to the discriminant hypersurface, endowed with the monodromy action of the associated Milnor fibre. The cohomology of G with trivial coefficients is derived from the previous one. | The K(π, 1) problem for the affine Artin group of type \widetilde{B}_n
and its cohomology | 12,841 |
Physicists showed that the generating function of orbifold elliptic genera of symmetric orbifolds can be written as an infinite product. We show that there exists a geometric factorization on space level behind this infinite product formula in much more general framework, where factors in the infinite product correspond to isomorphism classes of connected finite covering spaces of manifolds involved. From this formula, a concept of geometric Hecke operators for functors emerges. We show that these Hecke operators indeed satisfy the usual identity of Hecke operators for the case of 2-dimensional tori. | Infinite Product Decomposition of Orbifold Mapping Spaces | 12,842 |
Chas and Sullivan showed that the homology of the free loop space LM of an oriented closed smooth finite dimensional manifold M admits the structure of a Batalin-Vilkovisky (BV) algebra equipped with an associative product called the loop product and a Lie bracket called the loop bracket. We show that the cap product is compatible with the above two products in the loop homology. Namely, the cap product with cohomology classes coming from M via the circle action acts as derivations on loop products as well as on loop brackets. We show that Poisson identities and Jacobi identities hold for the cap product action, extending the BV structure in the loop homology to the one including the cohomology of M. Finally, we describe the cap product in terms of the BV algebra structure in the loop homology. | Cap Products in String Topology | 12,843 |
Cohen and Godin constructed positive boundary topological quantum field theory (TQFT) structure on the homology of free loop spaces of oriented closed smooth manifolds by associating a certain operations called string operations to orientable surfaces with parametrized boundaries. We show that all TQFT string operations associated to surfaces of genus at least one vanish identically. This is a simple consequence of properties of the loop coproduct which will be discussed in detail. One interesting property is that the loop coproduct is nontrivial only on the degree $d$ homology group of the connected component of $LM$ consisting of contractible loops, where $d=\dim M$, with values in the degree 0 homology group of constant loops. Thus the loop coproduct behaves in a dramatically simpler way than the loop product. | Loop coproducts in string topology and triviality of higher genus TQFT
operations | 12,844 |
Chas and Sullivan proved the existence of a Batalin-Vilkovisky algebra structure in the homology of free loop spaces on closed finite dimensional smooth manifolds using chains and chain homotopies. This algebraic structure involves an associative product called the loop product, a Lie bracket called the loop bracket, and a square 0 operator called the BV operator. Cohen and Jones gave a homotopy theoretic description of the loop product in terms of spectra. In this paper, we give an explicit homotopy theoretic description of the loop bracket and, using this description, we give a homological proof of the BV identity connecting the loop product, the loop bracket, and the BV operator. The proof is based on an observation that the loop bracket and the BV derivation are given by the same cycle in the free loop space, except that they differ by parametrization of loops. | A Homotopy Theoretic Proof of the BV Identity in Loop Homology | 12,845 |
We determine a family of functors from a poset to abelian groups such that the higher direct limits vanish on them. This is done by first characterizing the projective functors. Then a spectral sequence arising from the grading of the poset is used. Also the dual version for injective functors and higher inverse limits is included. Graded posets include simplicial complexes, subdivision categories and simplex-like posets. | A family of acyclic functors | 12,846 |
We present a method to compute integral cohomology of posets. This toolbox is applicable as soon as the sub-posets under each object possess certain structure. This is the case for simplicial complexes and simplex-like posets. The method is based on homological algebra arguments in the category of functors and on a spectral sequence built upon the poset. We show its relation to discrete Morse theory. As application we give an alternative proof of Webb's conjecture for saturated fusion systems and we compute the cohomology of Coxeter complexes for finite and infinite Coxeter groups. | A method for integral cohomology of posets | 12,847 |
We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L_{K(2)S^0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E_2^hF where F is a finite subgroup of the Morava stabilizer group and E_2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n=2 at p=3 represents the edge of our current knowledge: n=1 is classical and at n=2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic. | A resolution of the K(2)-local sphere at the prime 3 | 12,848 |
We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these modules, and present several examples. We also provide a geometric interpretation for the higher tor-modules of homology modules of multi-filtered simplicial complexes. | A refinement of multi-dimensional persistence | 12,849 |
We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of the loop space of X. This gives a homotopy-theoretic version of the correspondence between covering spaces over X and sets with an action of the fundamental group of X. We then use these two equivalences to study base change functors for parametrized spaces. | Parametrized spaces model locally constant homotopy sheaves | 12,850 |
Let $X$ be a nilpotent space such that there exists $N\geq 1$ with $H^N(X,\mathbb Q) \ne 0$ and $H^n(X,\mathbb Q)=0$ if $n>N$. Let $Y$ be a m-connected space with $m\geq N+1$ and $H^*(Y,\mathbb Q)$ is finitely generated as algebra. We assume that the odd part of the rational Hurewicz homomorphism: $\pi_{odd}(X)\otimes \mathbb Q\to H_{odd}(X,\mathbb Q)$ is non-zero. We prove that if the space $\mathcal F(X,Y)$ of continuous maps from $X$ to $Y$ is rationally formal, then $Y$ has the rational homotopy type of a finite product of Eilenberg Mac Lane spaces. At the opposite, we exhibit an example of a rationally formal space $\mathcal F(S^2,Y)$ where $Y$ is not rationally equivalent to a product of Eilenberg Mac Lane spaces. | Rational formality of function spaces | 12,851 |
A classical theorem of H. Hopf asserts that a closed connected smooth manifold admits a nowhere vanishing vector field if and only if its Euler characteristic is zero. R. Brown generalized Hopf's result to topological manifolds, replacing vector fields with path fields. In this note, we give an equivariant analog of Brown's theorem for locally smooth $G$-manifolds where $G$ is a finite group. | Equivariant path fields on topological manifolds | 12,852 |
On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration. | Diagonal fibrations are pointwise fibrations | 12,853 |
Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes $F_n$ are constructed. An explicit diagonal on $F_n$ is defined and a multiplicative model for the free loop fibration $\Omega Y\to \Lambda Y\to Y$ is obtained. As an application we establish an algebra isomorphism $H^*(\Lambda Y;\mathbb{Z}) \approx S(U)\otimes \Lambda(s^{_{-1}}U)$ for the polynomial cohomology algebra $H^*(Y;\mathbb{Z})=S(U).$ | The bitwisted Cartesian model for the free loop fibration | 12,854 |
We prove two results from Morita theory of stable model categories. Both can be regarded as topological versions of recent algebraic theorems. One is on recollements of triangulated categories, which have been studied in the algebraic case by J{\o}rgensen. We give a criterion which answers the following question: When is there a recollement for the derived category of a given symmetric ring spectrum in terms of two other symmetric ring spectra? The other result is on well generated triangulated categories in the sense of Neeman. Porta characterizes the algebraic well generated categories as localizations of derived categories of DG categories. We prove a topological analogon: a topological triangulated category is well generated if and only if it is triangulated equivalent to a localization of the derived category of a symmetric ring spectrum with several objects. Here `topological' means triangulated equivalent to the homotopy category of a spectral model category. Moreover, we show that every well generated spectral model category is Quillen equivalent to a Bousfield localization of a category of modules via a single Quillen functor. | Two results from Morita theory of stable model categories | 12,855 |
In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors. | Brown representability for space-valued functors | 12,856 |
Due to the deep work of Tillmann, Madsen, Weiss and Galatius, the cohomology of the stable mapping class group $\gaminf$ is known with rational or finite field coefficients. Little is known about the integral cohomology. In this paper, we study the first four cohomology groups. Also, we compute the first few steps of the Postnikov tower of $B \gaminf^+$, the Quillen plus construction applied to $B \gaminf$. Our method relies on the Madsen-Weiss theorem, a few known computations of stable homotopy groups of spheres and projective spaces and on a certain action of the binary icosahedral group on a surface. Using the latter, we can also describe an explicit geometric generator of the third homotopy group $\pi_3 (B \gaminf)$. | The low-dimensional homotopy of the stable mapping class group | 12,857 |
We study a correspondence between orientation reversing involutions on compact 3-manifolds with only isolated fixed points and binary, self-dual codes. We show in particular that every such code can be obtained from such an involution. We further relate doubly even codes to Pin^- -structures and Spin-manifolds. | Involutions on 3-Manifolds and Self-dual, Binary Codes | 12,858 |
We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of spaces and small categories, by using partially ordered sets. This yields a new conceptual proof to the well-known fact that these two homotopy categories are equivalent. | On the subdivision of small categories | 12,859 |
Motivated by the cohomology theory of loop spaces, we consider a special class of higher order homotopy commutative differential graded algebras and construct the filtered Hirsch model for such an algebra $A$. When $x\in H(A)$ with $\mathbb{Z}$ coefficients and $x^{2}=0,$ the symmetric Massey products $% \langle x\rangle ^{n}$ with $n\geq 3$ have a finite order (whenever defined). However, if $\Bbbk $ is a field of characteristic zero, $\langle x\rangle ^{n}$ is defined and vanishes in $H(A\otimes \Bbbk )$ for all $n$. If $p$ is an odd prime, the Kraines formula $\langle x\rangle ^{p}=-\beta \mathcal{P}_{1}(x)$ lifts to $H^{\ast }(A\otimes {\mathbb{Z}}_{p}).$ Applications of the existence of polynomial generators in the loop homology and the Hochschild cohomology with a $G$-algebra structure are given. | Filtered Hirsch Algebras | 12,860 |
Let M be a simply-connected closed manifold and consider the (ordered) configuration space of $k$ points in M, F(M,k). In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F(M,k). We prove that our model it is at least a Sigma_k-equivariant differential graded model. We also study Lefschetz duality at the level of cochains and describe equivariant models of the complement of a union of polyhedra in a closed manifold. | A remarkable DG-module model for configuration spaces | 12,861 |
For smooth manifolds $M$ and $N$, let $\Ebar(M, N)$ be the homotopy fiber of the map $\Emb(M, N)\longrightarrow \Imm(M, N)$. Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula $V\mapsto \Sigma^\infty\Ebar(M, N\times V)$. In this paper, we describe the Taylor polynomials of this functor, in the sense of M. Weiss' orthogonal calculus, in the case when $N$ is a nice open submanifold of a Euclidean space. This leads to a description of the derivatives of this functor when $N$ is a tame stably parallelizable manifold (we believe that the parallelizability assumption is not essential). Our construction involves a certain space of rooted forests (or, equivalently, a space of partitions) with leaves marked by points in $M$, and a certain ``homotopy bundle of spectra'' over this space of trees. The $n$-th derivative is then described as the ``spectrum of restricted sections'' of this bundle. This is the first in a series of two papers. In the second part, we will give an analogous description of the derivatives of the functor $\Ebar(M, N\times V)$, involving a similar construction with certain spaces of connected graphs (instead of forests) with points marked in $M$. | Derivatives of embedding functors I: the stable case | 12,862 |
In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence between its localization with respect to weak equivalences and the localised category of cofibrant objets with respect to strong equivalences. This equivalence allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and functor categories with a triple, in the last case we find examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications, we prove the existence of filtered minimal models for \emph{cdg} algebras over a zero-characteristic field and we formulate an acyclic models theorem for non additive functors. | A Cartan-Eilenberg approach to Homotopical Algebra | 12,863 |
In this note, we define the notion of a cactus set, and show that its geometric realization is naturally an algebra over Voronov's cactus operad, which is equivalent to the framed 2-dimensional little disks operad $\mathcal{D}_2$. Using this, we show that the Hochschild cohomology of a Poincar\'e algebra A is an algebra over (the chain complexes of) $\mathcal{D}_2$. | The Hochschild cohomology of a Poincaré algebra | 12,864 |
For a Whitney stratification S of a space X (or more generally a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category $EP_{\leq 2}(X,S)$, called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors from $EP_{\leq 2}(X,S)$ to the 2-category of small categories. | Exit paths and constructible stacks | 12,865 |
A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology. | An Algebraic Chain Model of String Topology | 12,866 |
This correction article is actually unnecessary. The proof of Theorem 1.2, concerning commutative HQ-algebra spectra and commutative differential graded algebras, in the author's paper [American Journal of Mathematics vol. 129 (2007) 351-379 (arxiv:math/0209215v4)] is correct as originally stated. Neil Strickland carefully proved that D is symmetric monoidal; so Proposition 4.7 and hence also Theorem 1.2 hold as stated. Strickland's proof will appear in joint work with Stefan Schwede; see related work in Strickland's [arxiv:0810.1747]. Note here D is defined as a colimit of chain complexes; in contrast, non-symmetric monoidal functors analogous to D are defined as homotopy colimits of spaces in previous work of the author. | Correction to: HZ-algebra spectra are differential graded algebras | 12,867 |
In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed simplicial groups and the homological algebra of module-valued functors. The symmetric homology of group algebras is related to stable homotopy theory. Two spectral sequences for computing symmetric homology are constructed. The relation to cyclic homology is discussed and some conjectures and questions towards further work are discussed. | Symmetric Homology of Algebras | 12,868 |
Given CW complexes X and Y, let map(X,Y) denote the space of continuous functions from X to Y with the compact open topology. The space map(X,Y) need not have the homotopy type of a CW complex. Here the results of an extensive investigation of various necessary and various sufficient conditions for map(X,Y) to have the homotopy type of a CW complex are exhibited. The results extend all previously known results on this topic. Moreover, appropriate converses are given for the previously known sufficient conditions. It is shown that this difficult question is related to well known problems in algebraic topology. For example, the geometric Moore conjecture (asserting that a simply connected finite complex admits an eventual geometric exponent at any prime if and only if it is elliptic) can be restated in terms of CW homotopy type of certain function spaces. Spaces of maps between CW complexes are a particular case of inverse limits of systems whose bonds are Hurewicz fibrations between spaces of CW homotopy type. Related problems concerning CW homotopy type of the limit space of such a system are also studied. In particular, an almost complete solution to a well known problem concerning towers of fibrations is presented. | CW type of inverse limits and function spaces | 12,869 |
In 1996, Jens Franke proved the equivalence of certain triangulated categories possessing an Adams spectral sequence. One particular application of this theorem is that the K_(p)-local stable homotopy category at an odd prime can be described as the derived category of an abelian category. We explain this proof from a topologist's point of view. | On the algebraic classification of K-local spectra | 12,870 |
In this paper, we study $Z_2$ actions on a cell complex X having the cohomology ring isomorphic to that of the wedge sum $P^2 (n) V S^{3n}$ or $S^n V S^{2n} V S^{3n}$. We determine the possible fixed point sets depending on whether or not X is totally non-homologous to zero in $X_{Z_2}$ and give examples realizing the possible cases. | Z_2 actions on complexes with three non-trivial cells | 12,871 |
The notions of deleted and restricted arrangements have been useful in the study of arrangements of hyperplanes. If A is an arrangement of hyperplanes, x in A and A', A'' the deleted and restricted arrangements, there is a formula connecting the Poincare polynomials of the complement spaces M(A), M(A') and M(A''). In this paper, we consider the extension of this formula to arbitrary subspaces arrangements. The main result is the existence of a long exact sequence connecting the rational cohomology of M(A), M(A') and M(A''). Using this sequence, we obtain new results connecting the Betti numbers and Poincare polynomials of deleted and restricted arrangements. | A long exact sequence in cohomology for deleted and restricted subspaces
arrangements | 12,872 |
The aim of this article is to give a criterion, generalizing the criterion introduced by Priddy for algebras, to verify that an operad is Koszul. We define the notion of a Poincare-Birkhoff-Witt basis in the context of operads. Then we show that an operad having a Poincare-Birkhoff-Witt basis is Koszul. Besides, we obtain that the Koszul dual operad has also a Poincare-Birkhoff-Witt basis. We check that the classical examples of Koszul operads (commutative, associative, Lie) have a Poincare-Birkhoff-Witt basis. | A Poincaré-Birkhoff-Witt criterion for Koszul operads | 12,873 |
We study the notion of twisting elements $da=a\cup_1a$ with respect to $\cup_1$ product when it is a part of homotopy Gerstenhaber algebra structure. This allows to bring to one context the two classical concepts, the theory of deformation of algebras of M. Gerstenhaber, and $A(\infty)$-algebras of J. Stasheff. | Twisting Elements in Homotopy G-algebras | 12,874 |
Let $G = \ZZ_p$, $p$ an odd prime, act freely on a finite-dimensional CW-complex $X$ with mod $p$ cohomology 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 $X/G$, when $p^2\nmid m$. | On the cohomology of orbit space of free \pmb{${\ZZ}_{p}$}-actions on
lens spaces | 12,875 |
The search for higher homotopy Hopf algebras (known today as A_\infty-bialgebras) began in 1996 during a conference at Vassar College honoring Jim Stasheff in the year of his 60th birthday. In a talk entitled "In Search of Higher Homotopy Hopf Algebras", I indicated that a DG Hopf algebra could be thought of as some (unknown) higher homotopy structure with trivial higher order structure and deformed using a graded version of Gerstenhaber and Schack's bialgebra deformation theory. In retrospect, the bi(co)module structure encoded in Gerstenhaber and Schack's differential defining deformation cohomology detects some (but not all) of the A_infty-bialgebra structure relations. Nevertheless, this motivated the discovery of A_infty-bialgebras by S. Saneblidze and myself in 2005. | Higher Homotopy Hopf Algebras Found: A Ten Year Retrospective | 12,876 |
We determine the v1-periodic homotopy groups of all irreducible p-compact groups (BX,X). In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, if p is odd, every irreducible p-compact group has X of the homotopy type of a product of explicit spaces related to p-completed Lie groups. | Homotopy type and v1-periodic homotopy groups of p-compact groups | 12,877 |
As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is Cartesian closed and that the forgetful functor to the category of compactly generated spaces creates all limits and colimits. | A convenient category of locally preordered spaces | 12,878 |
The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homotopic, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying space has connected CW type. | Criteria for homotopic maps to be so along monotone homotopies | 12,879 |
We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $\pi_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $\Omega F$ have values in homotopy n-nilpotent groups. | Homotopy nilpotent groups | 12,880 |
Cohen, Moore, and Neisendorfer's work on the odd primary homotopy theory of spheres and Moore spaces, as well as the first author's work on the secondary suspension, predicted the existence of a p-local fibration S^2n-1 --> T --> \Omega S^2n+1 whose connecting map is degree p^r. In a long and complex monograph, Anick constructed such a fibration for p>= 5 and r>= 1. Using new methods we give a much more conceptual construction which is also valid for p=3 and r>= 1. We go on to establish several properties of the space T. | An elementary construction of Anick's fibration | 12,881 |
The main result of the paper is a formula for the fundamental group of the coarse moduli space of a topological stack. As an application, we find simple general formulas for the fundamental group of the coarse quotient of a group action on a topological space in terms of the fixed point data. The formulas seem, surprisingly, to be new. In particular, we recover, and vastly generalize, results of Armstrong, Bass, Higgins, Rhodes. | Fundamental groups of topological stacks with slice property | 12,882 |
The paper gives a new proof that the model categories of stable modules for the rings Z/(p^2) and (Z/p)[\epsilon]/(\epsilon^2) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with non-isomorphic K-theories. | A curious example of two model categories and some associated
differential graded algebras | 12,883 |
This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category $\Tops$ of stratified spaces, that are topological spaces $X$ endowed with a partition $\cF$ and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element $(X,\cF)$ of $\Tops$ together with a class $\cA$ of subsets of $X$; they are similar to invariants introduced by M. Clapp and D. Puppe. If $(X,\cF)\in\Tops$, we define a transverse subset as a subspace $A$ of $X$ such that the intersection $S\cap A$ is at most countable for any $S\in \cF$. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a $C^1$-foliation, the three previous definitions, with $\cA$ the class of transverse subsets, coincide with the tangential category and are homotopical invariants. | Ganea and Whitehead definitions for the tangential
Lusternik-Schnirelmann category of foliations | 12,884 |
We consider the manifold $Fl_n(\mathbb{H})=Sp(n)/Sp(1)^n$ of all complete flags in $\mathbb{H}^n$, where $\mathbb{H}$ is the skew-field of quaternions. We study its equivariant $K$-theory rings with respect to the action of two groups: $Sp(1)^n$ and a certain canonical subgroup $T:=(S^1)^n\subset Sp(1)^n$ (a maximal torus). For the first group action we obtain a Goresky-Kottwitz-MacPherson type description. For the second one, we describe the ring $K_T(Fl_n(\mathbb{H}))$ as a subring of $K_T(Sp(n)/T)$. This ring is well known, since $Sp(n)/T$ is a complex flag variety. | Equivariant $K$-theory of quaternionic flag manifolds | 12,885 |
We determine the cup-length of some oriented Grassmann manifolds by finding a Groebner basis associated with a certain subring of the cohomology of them. As its applications, we provide not only a lower but also an upper bound for the LS-category of some oriented Grassmann manifolds. We also study the immersion problem of them. | Application of Groebner bases to the cup-length of oriented Grassmann
manifolds | 12,886 |
Let E_n be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E^{nr}_n whose coefficients are built from the coefficients of E_n and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E^{nr}_n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that there are no non-trivial connected Galois extensions of E^{nr}_n with Galois group a finite group G with cyclic quotient. Our results carry over to the K(n)-local context. | Galois extensions of Lubin-Tate spectra | 12,887 |
In this work, we compare the two approximations of a path-connected space $X$, by the Ganea spaces $G_n(X)$ and by the realizations $\|\Lambda_\bullet X\|_{n}$ of the truncated simplicial resolutions emerging from the loop-suspension cotriple $\Sigma\Omega$. For a simply connected space $X$, we construct maps $\|\Lambda_\bullet X\|_{n-1}\to G_n(X)\to \|\Lambda_\bullet X\|_{n}$ over $X$, up to homotopy. In the case $n=2$, we prove the existence of a map $G_2(X)\to\|\Lambda_\bullet X\|_{1}$ over $X$ (up to homotopy) and conjecture that this map exists for any $n$. | Simplicial resolutions and Ganea fibrations | 12,888 |
Let M(1) be the mod 2 Moore spectrum. J.F. Adams proved that M(1) admits a minimal v_1-self map v_1^4: Sigma^8 M(1) -> M(1). Let M(1,4) be the cofiber of this self-map. The purpose of this paper is to prove that M(1,4) admits a minimal v_2-self map of the form v_2^32: Sigma^192 M(1,4) -> M(1,4). The existence of this map implies the existence of many 192-periodic families of elements in the stable homotopy groups of spheres. | On the existence of a v_2^32-self map on M(1,4) at the prime 2 | 12,889 |
We prove a chain rule for the Goodwillie calculus of functors from spectra to spectra. We show that the (higher) derivatives of a composite functor $FG$ at a base object $X$ are given by taking the composition product (in the sense of symmetric sequences) of the derivatives of $F$ at $G(X)$ with the derivatives of $G$ at $X$. We also consider the question of finding $P_n(FG)$, and give an explicit formula for this when $F$ is homogeneous. | A chain rule for Goodwillie derivatives of functors from spectra to
spectra | 12,890 |
In this note we study umkehr maps in generalized (co)homology theories arising from the Pontrjagin-Thom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which any umkehr homomorphism must satisfy. We use a version of Brown representability to show that these axioms completely characterize these homomorphisms, and a resulting uniqueness theorem follows. Finally, motivated by constructions in string topology, we extend this axiomatic treatment of umkehr homomorphisms to a fiberwise setting. | Umkehr Maps | 12,891 |
Generalizing work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra $C$ is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex $\cohoch (C)$ admits a natural comultiplicative structure. In particular, if $K$ is a reduced simplicial set and $C_{*}K$ is its normalized chain complex, then $\cohoch (C_{*}K)$ is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on $\cohoch (C_{*}K)$ when $K$ is a simplicial suspension. The coHochschild complex construction is topologically relevant. Given two simplicial maps $g,h:K\to L$, where $K$ and $L$ are reduced, the homology of the coHochschild complex of $C_{*}L$ with coefficients in $C_{*}K$ is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of $g$ and $h$, and this isomorphism respects comultiplicative structure. In particular, there a isomorphism, respecting comultiplicative structure, from the homology of $\cohoch(C_{*}K)$ to $H_{*}\op L|K|$, the homology of the free loops on the geometric realization of $K$. | CoHochschild homology of chain coalgebras | 12,892 |
We determine the integral cohomology rings of an infinite family of p-groups, for odd primes p, with cyclic derived subgroups. Our method involves embedding the groups in a compact Lie group of dimension one, and was suggested by P H Kropholler and J Huebschmann. | The integral cohomology rings of some p-groups | 12,893 |
We determine the mod-p cohomology rings of an infinite family of p-groups, for odd primes p, with cyclic derived subgroups. Our method involves embedding the groups in a compact Lie group of dimension one, and was suggested by P. H. Kropholler and J. Huebschmann. | The mod-p cohomology rings of some p-groups | 12,894 |
For each odd prime p, we exhibit p-groups G of p-rank two such that (suitably defined) Chern classes of unitary representations of G fail to generate the following rings: 1. The even degree integral cohomology of G; 2. The final page of the Atiyah-Hirzebruch spectral sequence for G; 3. The Brown-Peterson generalized cohomology of G. It follows that these groups afford counterexamples to conjectures of C. B. Thomas, M. F. Atiyah and P. Landweber. | Some examples in the integral and Brown-Peterson cohomology of p-groups | 12,895 |
We consider the Lyndon-Hochschild-Serre spectral sequence with mod-p coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. For this spectral sequence the second and third differentials are known, and we give a description for the fourth differential. Using this result we deduce a similar formula for the Serre spectral sequence for a principal fibration with fibre the classifying space of a cyclic p-group. The differential from odd rows to even rows involves a Massey triple product, so we describe the calculation of such products in the cohomology of a finite abelian group. As an example we determine the Poincare series for the mod-3 cohomology of various 3-groups. Remarks. 1) My definition of the higher differentials $d_i$ for $i\geq 2$ in the spectral sequence for a double chain complex differs from the usual one by a factor of $(-1)^{i+1}$. Both conventions are consistent, but the usual definition has the advantage of agreeing with the ``obvious'' definition of the differentials in the spectral sequence for the associated filtered chain complex. All of the theorems in this paper remain true exactly as stated if the more usual definition of $d_i$ is taken. 2) Carles Broto found a small mistake in this paper: the result for fibrations with fibre the classifying space of a cyclic group is stated for arbitrary fibrations, although it is only proved for principal fibrations. Since it is apparent from the first sentence of the proof that only principal fibrations are being considered, I have not bothered to publish an erratum. | A differential in the Lyndon-Hochschild-Serre spectral sequence | 12,896 |
For each prime p, we exhibit pairs of p-groups all of whose integral cohomology groups are isomorphic. The method used involves very little calculation. The groups are exhibited as kernels of homomorphisms from a compact Lie group G to U(1), and the main result is that kernels of `similar' elements of Hom(G,U(1)) have isomorphic integral cohomology groups. The 2-groups constructed in this version have been corrected (there was a mistake in the presentations given in the published paper). | p-Groups are not determined by their integral cohomology groups | 12,897 |
We give a lower bound for the exponent of certain elements in the integral cohomology of the total spaces of principal BC-bundles for C a finite cyclic group. As applications we give a proof of the theorem of A. Adem and H.-W. Henn that a p-group is elementary abelian if and only if its integral cohomology has exponent p, and we exhibit some infinite groups of finite virtual cohomological dimension whose Tate-Farrell cohomology contains torsion of order greater than the l.c.m. of the orders of their finite subgroups. We also give an upper bound for the exponent of all but finitely many of the integral cohomology groups of a finite group, in terms of the permutation representations of the group. | A bound on the exponent of the cohomology of BC-bundles | 12,898 |
We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses) for the automorphism group of V. We use Brauer characters and computer calculations. We construct and implement an algorithm for finding permutation submodules of maximal dimension inside modules for p-groups in characteristic p. | On the GL(V)-module structure of K(n)^*(BV) | 12,899 |
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