Crossed modules as homotopy normal maps

In this note we consider crossed modules of groups (N→G, G→Aut(N)), as a homotopy version of the inclusion N⊂G of a normal subgroup. Our main observation is a characterization of the underlying map N→G of a crossed module in terms of a simplicial group structure on the associated bar construction. This approach allows for “natural” generalizations to other monoidal categories, in particular we consider briefly what we call “normal maps” between simplicial groups.     

Homotopy classification of graded Picard categories

Certain low-dimensional symmetric cohomology groups of G-modules, for any given group G, are computed as the cohomology of an explicit cochain complex. This result is used to establish natural one-to-one correspondences between elements of the 3rd symmetric cohomology groups of G-modules, G-equivariant pointed 2-connected homotopy 4-types, and equivalence classes of G-graded Picard categories. The simplicial nerve of a G-graded Picard category is also constructed and studied.     

The dualizing spectrum of a topological group

To a topological group G, we assign a naive G-spectrum , called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that detects Poincaré duality in the sense that BG is a Poincaré duality space if and only if is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a norm map which is defined for any G and for any naive G-spectrum E. Applications of the dualizing spectrum come in two flavors: (i) applications in the theory of Poincaré duality spaces, and (ii) applications in the theory of group cohomology. On the Poincaré duality space side, we derive a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of fini the total space satisfies Poincaré duality if and only if the base and fiber do. The dualizing spectrum can also be used to give an entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincaré duality space. We also include a new proof of Browder's theorem that every finite H-space satisfies Poincaré duality. In connection with group cohomology, we show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. When E is connective, and when G admits a subgroup H of finite index such that BH is finitely dominated, we show that this cohomology coincides with the ordinary cohomology of G with coefficients in E in degrees greater than the cohomological dimension of H. In an appendix, we identify the homotopy type of for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. Received July 14, 1999 / Revised May 17, 2000 / Published online February 5, 2001     

A stable approach to the equivariant Hopf theorem

Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner [S. Waner, Equivariant RO(G)-graded bordism theories, Topology and its Applications 17 (1984) 1-26], the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterised in terms of congruences. This is first shown to be a stable problem, and then solved using methods of equivariant stable homotopy theory with respect to a semi-free G-universe.     

Equivariant homotopical homology with coefficients in a Mackey functor

Let M be a Mackey functor for a finite group G. In this paper, generalizing the Dold-Thom construction, we construct an ordinary equivariant homotopical homology theory with coefficients in M, whose values on the category of finite G-sets realize the bifunctor M, both covariantly and contravariantly. Furthermore, we extend the contravariant functor to define a transfer in the theory for G-equivariant covering maps. This transfer is given by a continuous homomorphism between topological abelian groups.We prove a formula for the composite of the transfer and the projection of a G-equivariant covering map and characterize those Mackey functors M for which that formula has an expression analogous to the classical one.     

Rational homotopy of the (homotopy) fixed point sets of circle actions

In this paper, when G is the circle S¹ and M is a G-space, we study the rational homotopy type of the fixed point set MG, the homotopy fixed point set MhG, and the natural injection MG→MhG.     

Simple homotopy types and finite spaces

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X↘Y of finite spaces induces a simplicial collapse K(X)↘K(Y) of their associated simplicial complexes. Moreover, a simplicial collapse K↘L induces a collapse X(K)↘X(L) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. This class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.     

Examples of cohomology manifolds which are not homologically locally connected

Bredon has constructed a 2-dimensional compact cohomology manifold which is not homologically locally connected, with respect to the singular homology. In the present paper we construct infinitely many such examples (which are in addition metrizable spaces) in all remaining dimensions n?3.     

On the algebras over equivariant little disks

We describe the structure present in algebras over the little disks operads for various representations of a finite group G, including those that are not necessarily universe or that do not contain trivial summands. We then spell out in more detail what happens for $G=C_2$, describing the structure on algebras over the little disks operad for the sign representation. Here we can also describe the resulting structure in Bredon homology. Finally, we produce a stable splitting of coinduced spaces analogous to the stable splitting of the product, and we use this to determine the homology of the signed James construction.     

CoHochschild homology of chain coalgebras

Generalizing the 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 admits a natural comultiplicative structure. In particular, if K is a reduced simplicial set and C_∗K is its normalized chain complex, then is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on when K is a simplicial suspension.The coHochschild complex construction is topologically relevant. Given two simplicial maps g,h:K→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 is an isomorphism, respecting comultiplicative structure, from the homology of to H_∗L|K|, the homology of the free loops on the geometric realization of K.     

Laminated decompositions involving a given submanifold

Let M denote a connected (n+1)-manifold (without boundary). We study laminated decompositions of M, by which we mean upper semicontinous decompositions G of M into closed, connected n-manifolds. In particular, given M with a lamination G and N, a locally flat, closed, n-dimensional submanifold, we determine conditions under which M admits another lamination G_N with N?G_N. For n ≠ 3 a sufficient condition is that i: N → M be a homotopy equivalence. For n > 3 we give examples to show that i: N → M being a homology equivalence is not sufficient. We also show how to replace the assumption of local flatness of N with a weaker cellularity criterion (n ? 4) known as the inessential loops condition. We then give examples illustrating the abundance of pathology if M is not assumed to have a preexisting lamination.     

Homotopy classification of braided graded categorical groups

For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises from the homotopy theory of G-spaces and it is called the “abelian cohomology of G-modules”. Then, as the main results of this paper, natural one-to-one correspondences between elements of the 3^rd cohomology groups of G-modules, G-equivariant pointed simply-connected homotopy 3-types and equivalence classes of braided G-graded categorical groups are established. The relationship among all these objects with equivariant quadratic functions between G-modules is also discussed.     

Representability theorems for presheaves of spectra

Suppose that M is a simplicial model category and that F is a contravariant simplicial functor defined on M which takes values in pointed simplicial sets. This note displays conditions on the simplicial model category M and the functor F such that F is representable up to weak equivalence. The conditions on F are homotopy coherent versions of the classical conditions for Brown representability, while M should have the fundamental properties of the stable model structure for presheaves of spectra on a Grothendieck site.     

Self homotopy groups with large nilpotency classes

We demonstrate that for any n>0 there exists a compact connected Lie group G such that the self homotopy group [G,G] has the nilpotency class greater than n, where [G,G] is a nilpotent group for a compact connected Lie group G.     

Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM on M, Witten defines an inhomogeneous coboundary operator dXM=d+ιXM on invariant forms on M. The main purpose is to adapt Belishev-Sharafutdinov?s boundary data to invariant forms in terms of the operator dXM in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator ΛXM on invariant forms on the boundary which we call the XM-DN map and using this we recover the XM-cohomology groups from the generalized boundary data (∂M,ΛXM). This shows that for a Zariski-open subset of the Lie algebra, ΛXM determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of XM-cohomology groups from ΛXM. These results explain to what extent the equivariant topology of the manifold in question is determined by ΛXM.     

Splitting manifolds as M × R where M has a k-fold end structure

Let X be a locally finite simplicial complex of dimension n, n? 5, equipped with a k-fold end structure [4] and consider a piecewise linear (n + 1)-dimensional manifold M that is proper homotopy equivalent to X × R by F:M → X × R, where R is the set of real numbers. The question arises as to whether or not the manifold M can be split, i.e., written as M = N × R where N is a n-manifold and where there is a proper homotopy between F and (p₁ ° F₀) × id:N × R → X × R, preserving the natural (k+1)-fold end structure, where F₀ is F|_N and p₁ is the projection X × R → X. Of particular significance is the fact that X is noncompact. When the construction of such splittings is attempted, algebraic obstructions arise, which vanish if and only if the construction can be completed. This paper develops such an obstruction theory by utilizing methods of L.C. Siebenmann and the k-fold end structures of F. Waldhausen.     

G-symmetric spectra,semistability and the multiplicative norm

In this paper we develop the basic homotopy theory of G-symmetric spectra (that is, symmetric spectra with a G-action) for a finite group G, as a model for equivariant stable homotopy with respect to a G-set universe. This model lies in between Mandell's equivariant symmetric spectra and the G-orthogonal spectra of Mandell and May and is Quillen equivalent to the two. We further discuss equivariant semistability, construct model structures on module, algebra and commutative algebra categories and describe the homotopical properties of the multiplicative norm in this context.     

On string topology of classifying spaces

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.     

On the homotopy type of CW-complexes with aspherical fundamental group

This paper is concerned with the homotopy type distinction of finite CW-complexes. A (G,n)-complex is a finite n-dimensional CW-complex with fundamental-group G and vanishing higher homotopy-groups up to dimension n−1. In case G is an n-dimensional group there is a unique (up to homotopy) (G,n)-complex on the minimal Euler-characteristic level χmin(G,n). For every n we give examples of n-dimensional groups G for which there exist homotopically distinct (G,n)-complexes on the level χmin(G,n)+1. In the case where n=2 these examples are algebraic.