Mathematics

In 1962, Wall showed that smooth, closed, oriented, (n−1) -connected 2n -manifolds of dimension at least 6 are classified up to connected sum with an exotic sphere by an algebraic refinement of the intersection form which he called an n -space. In this paper, we complete the determination of which n -spaces are realizable by smooth, closed, oriented, (n−1) -connected 2n -manifolds for all n≠63 . In dimension 126 the Kervaire invariant one problem remains open. Along the way, we completely resolve conjectures of Galatius-Randal-Williams and Bowden-Crowley-Stipsicz, showing that they are true outside of the exceptional dimension 23 , where we provide a counterexample. This counterexample is related to the Witten genus and its refinement to a map of E ∞ -ring spectra by Ando-Hopkins-Rezk. By previous work of many authors, including Wall, Schultz, Stolz and Hill-Hopkins-Ravenel, as well as recent joint work of Hahn with the authors, these questions have been resolved for all but finitely many dimensions, and the contribution of this paper is to fill in these gaps.

Let us consider the prime field of two elements, F 2 . One of the open problems in Algebraic topology is the hit problem for a module over the mod 2 Steenrod algebra A . The problem asks a minimal set of generators for the polynomial algebra P m := F 2 [ x 1 , x 2 ,?? x m ] regarded as a connected unstable A -module on m variables x 1 ,?? x m , each of degree one. The algebra P m is the cohomology with F 2 -coefficients of the product of m copies of the Eilenberg-MacLan complex K( F 2 ,1). The hit problem has been thoroughly studied for 35 years in a variety of contexts, but it remains open for m??. The aim of this work is of studying the hit problem of five variables. An efficient approach for solving the problem has been given. More precisely, we develop our work in \cite{D.P3} on the hit problem for A -module P 5 in the "generic" degree n t :=5( 2 t ??)+ 18.2 t with t??. This result confirms Sum's conjecture \cite{N.S2} for the relation between the minimal sets of A -generators of the polynomial algebras P m?? and P m in the case m=5 and degree n t . Two applications of this study are to determine the dimension of the graded space F 2 ??A P 6 in the generic degree 5( 2 t+4 ??)+ n 1 .2 t+4 for all t>0 and modular representation of the general linear group of rank 5 over F 2 . As a corollary, we show that the Singer cohomological "transfer" is an isomorphism in bidegree (5,5+ n 0 ).

The independence complex Ind(G) of a graph G is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of Ind(G) that gives rise to a double complex with trivial homology. Filtering this double complex in the right direction induces a spectral sequence that converges to zero and contains on its first page the homology of the independence complexes of G and various subgraphs of G , obtained by removing independent sets and their neighborhoods from G . It is shown that this spectral sequence may be used to study the homology of Ind(G) . Furthermore, a careful investigation of the sequence's first page exhibits a relation between the cardinality of maximal independent sets in G and the vanishing of certain homology groups of the independence complexes of some subgraphs of G . This relation is shown to hold for all paths and cyclic graphs.

This paper studies the homology and cohomology of the Temperley-Lieb algebra TL_n(a), interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a=v+v^{-1} for some unit v in the ground ring, and states that the homology and cohomology vanish up to and including degree (n-2). To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when n is even. Our methods are inspired by the tools and techniques of homological stability for families of groups. We construct and exploit a chain complex of 'planar injective words' that is analogous to the complex of injective words used to prove stability for the symmetric groups. However, in this algebraic setting we encounter a novel difficulty: TL_n(a) is not flat over TL_m(a) for m<n, so that Shapiro's lemma is unavailable. We resolve this difficulty by constructing what we call 'inductive resolutions' of the relevant modules. Vanishing results for the homology and cohomology of Temperley-Lieb algebras can also be obtained from existence of the Jones-Wenzl projector. Our own vanishing results are in general far stronger than these, but in a restricted case we are able to obtain additional vanishing results via existence of the Jones-Wenzl projector. We believe that these results, together with the second author's work on Iwahori-Hecke algebras, are the first time the techniques of homological stability have been applied to algebras that are not group algebras.

We build model structures on the category of equivariant simplicial operads with weak equivalences determined by families of subgroups, in the context of operads with a varying set of colors (and building on the fixed color model structures in the prequel). In particular, by specifying to the family of graph subgroups (or, more generally, one of the indexing systems of Blumberg-Hill), we obtain model structures on the category of equivariant simplicial operads whose weak equivalences are determined by norm map data.

We show that quadratic and symmetric L-theory of the integers are related by Anderson duality and show that both spectra split integrally into the L-theory of the real numbers and a generalised Eilenberg-Mac Lane spectrum. As a consequence, we obtain a corresponding splitting of the space G/Top. Finally, we prove analogous results for the genuine L-spectra recently devised for the study of Grothendieck--Witt theory.

Real Bruhat cells give an important and well studied stratification of such spaces as G L n+1 , Fla g n+1 =S L n+1 /B , S O n+1 and Spi n n+1 . We study the intersections of a top dimensional cell with another cell (for another basis). Such an intersection is naturally identified with a subset of the lower nilpotent group L o 1 n+1 . We are particularly interested in the homotopy type of such intersections. In this paper we define a stratification of such intersections. As a consequence, we obtain a finite CW complex which is homotopically equivalent to the intersection. We compute the homotopy type for several examples. It turns out that for n?? all connected components of such subsets of L o 1 n+1 are contractible: we prove this by explicitly constructing the corresponding CW complexes. Conversely, for n?? and the top permutation, there is always a connected component with even Euler characteristic, and therefore not contractible. This follows from formulas for the number of cells per dimension of the corresponding CW complex. For instance, for the top permutation S 6 , there exists a connected component with Euler characteristic equal to 2 . We also give an example of a permutation in S 6 for which there exists a connected component which is homotopically equivalent to the circle S 1 .

We present the length, a numerical cohomological index theory, of G -spaces which are cohomology spheres and G is a p -torus or a torus group, where p is a prime. As a consequence, we obtain Borsuk-Ulam and Bourgin-Yang type theorems in this context. A sharper version of the Bourgin-Yang theorem for topological manifolds is also proved. Also, we give some general results regarding the upper and lower bound for the length.

The present note presents some results about the mod-2 cohomology, modulo nilpotent elements elements of the fiber E of a decomposable map ψ : K(Z, 2) → K(Z/2, p). This is more an announcement and a brief description of the tools that are used: Lannes' T functor and the Eilenberg-Moore spectral sequence.

We define a cup product on the cochain complex of a multisimplicial set, that is compatible with the classical cup product on the cochain complex of the diagonal simplicial set via the Eilenberg-Zilber map. This helps to speed up cochain level computations for multisimplicial complexes.