Stable Splittings Research Articles

Global stable splittings of Stiefel manifolds

We prove global equivariant refinements of Miller's stable splittings of the infinite orthogonal, unitary and symplectic groups, and more generally of the spaces $O/O(m)$, $U/U(m)$ and $Sp/Sp(m)$. As such, our results encode compatible equivariant stable splittings, for all compact Lie groups, of specific equivariant refinements of these spaces. In the unitary and symplectic case, we also take the actions of the Galois groups into account. To properly formulate these Galois-global statements, we introduce a generalization of global stable homotopy theory in the presence of an extrinsic action of an additional topological group.

Categorifying induction formulae via divergent series

We show how to get explicit induction formulae for finite group representations, and more generally for rational Green functors, by summing a divergent series over Dwyer's subgroup and centralizer decomposition spaces. This results in formulae with rational coefficients. The former space yields a well-known induction formula, the latter yields a new one. As essentially immediate corollaries of the existing literature, we get similar formulae in group cohomology and stable splittings of classifying spaces.

Stable splittings of Hilbert spaces of functions of infinitely many variables

We present an approach to defining Hilbert spaces of functions depending on infinitely many variables or parameters, with emphasis on a weighted tensor product construction based on stable space splittings. The construction has been used in an exemplary way for guiding dimension- and scale-adaptive algorithms in application areas such as statistical learning theory, reduced order modeling, and information-based complexity. We prove results on compact embeddings, norm equivalences, and the estimation of ϵ-dimensions. A new condition for the equivalence of weighted ANOVA and anchored norms is also given.

Representations of the double Burnside algebra and cohomology of the extraspecial p-group II

Let E be the extraspecial p-group of order p3 and exponent p where p is an odd prime. We determine the modp cohomology H⁎(X,Fp) of a summand X in the stable splitting of p-completed classifying space BE. In the previous paper (Hida and Yagita (2014) [7]), we determined this cohomology modulo nilpotence. In this paper, we consider the whole of the cohomology. Moreover, we consider the stable splittings of BG for some finite groups with Sylow p-subgroup E related to the three dimensional linear group L3(p).

Tubular configurations: equivariant scanning and splitting

Replacing configurations of points by configurations of tubular neighbourhoods (or discs) in a manifold M we are able to define a natural scanning map that is equivariant under the action of the diffeomorphism group of the manifold. We also construct the so-called power-set map of configuration spaces diffeomorphism equivariantly. Combining these two constructions yields stable splittings in the sense of Snaith and generalizations thereof that are equivariant. In particular, one deduces stable splittings of homotopy orbit spaces. As an application, the homology injectivity is proved for diffeomorphisms of M that fix an increasing number of points. Throughout, we work with configurations spaces with labels in a fibre bundle over M.

Homotopy decompositions of looped 2-cell co-H spaces

Let p be an odd prime. For certain p-local 2-cell co-H spaces Y, we consider the homotopy decomposition of ΩY. First we give an explicit description of the stable splitting of ΩY, which refines James' classical result on stable splittings of loop suspension. Also we show that the product decomposition of ΩY has as a factor ΩΣf(k)Y for f(k) a linear function for k≥0.

Saturated fusion systems as idempotents in the double Burnside ring

We give a new, unexpected characterization of saturated fusion systems on a p-group S in terms of idempotents in the p-local double Burnside ring of S that satisfy a Frobenius reciprocity relation, and reformulate fusion-theoretic phenomena in the language of idempotents. Interpreting our results in stable homotopy, we answer a long-standing question on stable splittings of classifying spaces of finite groups, and generalize the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. This work is partly motivated by a conjecture of Haynes Miller which proposes retractive transfer triples as a purely homotopy-theoretic model for p-local finite groups. We take an important step toward proving this conjecture by showing that a retractive transfer triple gives rise to a p-local finite group when two technical assumptions are made, thus reducing the conjecture to proving those two assumptions.

The tangent complex and Hochschild cohomology of -rings

AbstractIn this work, we study the deformation theory of${\mathcal {E}}_n$-rings and the${\mathcal {E}}_n$analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence$A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the${\mathcal {E}}_n$-tangent complex and${\mathcal {E}}_n$-Hochschild cohomology of an${\mathcal {E}}_n$-ring$A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups,$B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here${\mathfrak B}^n\!A$is an enriched$(\infty ,n)$-category constructed from$A$, and${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of${\mathfrak B}^n\!A$. These groups are associated to moduli problems in${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex$T_A[-n]$is a nonunital${\mathcal {E}}_{n+1}$-algebra. The${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known${\mathcal {E}}_{n+1}$-algebra structure on${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed$n$-manifolds with coefficients given by${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.

Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids

In this paper we generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition. The theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the Schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters. An assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small. We do not need to align them with possible discontinuities in the coefficients. The proofs make use of novel stable splittings based on weighted quasi-interpolants and weighted Poincare-type inequalities. Numerical experiments are...

Stable splittings, spaces of representations and almost commuting elements in Lie groups

AbstractIn this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one. By using symmetric products, the colimits Rep(ℤn, SU), Rep(ℤn, U) and Rep(ℤn, Sp) are explicitly described as finite products of Eilenberg–MacLane spaces.

Stable splittings of surface mapping spaces

We study the homotopy type of mapping spaces from Riemann surfaces to spheres. Our main result is a stable splitting of these spaces into a bouquet of new finite spectra. From this and classical results, one may deduce splittings of the configuration spaces of surfaces.

Stable splittings of classifying spaces of finite groups: Bridging the work of Benson-Feshbach and Martino-Priddy

We provide a direct connection between the work of David Benson and Mark Feshbach [2], and the work of John Martino and Stewart Priddy [9] on stable splittings of classifying spaces of finite groups.

An application of unstable K-theory

In [1] and [2], we introduced and investigated “unstable K-theory” $U_{n}(X) = [X,U(n)]$ and showed the relation with the Adams e-invariant. In this paper, we offer a theorem relating to $U_{n}(X)$ and show an application in connection with stable splittings of $G_{2}$.

Stable splittings of classifying spaces of 2-toral compact Lie groups

Let G be a compact Lie group and H be a p-toral subgroup of G. In this paper, we give a necessary and sufficient condition for a summand originating BH p to be a summand of BG p and also give the corresponding multiplicity for the compact Lie group G which is not even p-Roquette.

Endofiniteness in stable homotopy theory

We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite groups.

A generalization of Snaith-type filtration

In this paper we describe the Goodwillie tower of the stable homotopy of a space of maps from a finite-dimensional complex to a highly enough connected space. One way to view it is as a partial generalization of some wellknown results on stable splittings of mapping spaces in terms of configuration spaces. 0. INTRODUCTION It has been known for a while (see [1] for a survey article and a list of references) that given a parallelizable, compact m-dimensional manifold M with a nonempty boundary, and given a connected, pointed space Z, there is a configuration space model for the space of unbased maps Map(M, SmZ), which stably splits. More precisely, there is a weak equivalence: (0.1) Q? E' (Map(M, SmZ)) 1 QJJ E' ((C(M, OM; n) AEn ZAn)), n>1 where C(M, OM; n) stands for the space of n-tuples of distinct points in M, where all n-tuples whose intersection with OM is not empty have been identified to a point. There is an analogous splitting for the space of based maps. A closely related result is the stable splitting of spaces of the form QmEmX. This later splitting is sometimes refered to as the Snaith splitting (at least in the case m =oo), and we refer to (0.1) as Snaith-type splitting. It is, therefore, natural to ask if for a based space K, that is not a manifold, but, say, a finite CW-complex, anything can be said about the functor X F-+ QMap*(K, X) (where Map*(K, X) stands for the space of based maps from K to X). One does not expect this functor to split, in general, but it is still reasonable to try to approximate it by more elementary functors in a way that would give the splitting above in the case when K = M and X = SmZ. It turns out that this question (in fact a generalization of it) can be answered positively within the framework of the theory referred to as calculus of functors, which had been developed by T. Goodwillie in [4], [5], [6]. Since [6] has not been published yet, we present a brief outline of the theory of Taylor towers in the appendix. In what follows we will freely use notation from there. Consider again the identity (0.1). Observe that the factors Q (C(M, OM; n) AEn ZAn) Received by the editors July 21, 1994 and, in revised form, February 4, 1997. 1991 Mathematics Subject Classification. Primary 55P99. (?)1999 American Mathematical Society 1123 This content downloaded from 157.55.39.235 on Fri, 07 Oct 2016 05:53:45 UTC All use subject to http://about.jstor.org/terms

Induced maps in homology for p-toral compact Lie groups

For stable splittings of the classifying spaces of general p-toral compact Lie groups, it is important step to describe the induced maps of the stable maps on F p -homology. In this paper, we give the structure of the induced maps on F p -homology for the classifying spaces of p-toral compact Lie groups. For this purpose, we show that there exists a transfer map τ : BF ∞ p ∧ → BH ∞ p ∧ where F ∞ is a p-discrete toral group and H ∞ is a subgroup of F ∞ and we combine this result with the property that A f ( F ∞, K) is dense in lim ← nA(F n, K) p ∧ .

Stable splittings of classifying spaces of compact Lie group

Let G be a compact Lie gorup. In this paper, we study the stable splitting of BG completed at p into a wedge sum of indecomposable spectra. When G is finite, this question has been reduced to understanding the irreducible modular representations of the outer automorphism group Out (Q) for various p-subgroups Q ⊆ G by work of [2], [14] and [18]. The principal tool used by these authors is a generalization of Segal’s Burnside ring conjecture which describes all stable maps between p-completions of the classifying spaces of p-groups. One of the problems in going from finite groups to compact Lie groups is that in the latter case one no longer has a convenient description of all stable maps between classifying spaces. Our solution to this difficulty is to pass from the ring of stable self-maps to the induced self-maps on Fp-homology. It is well-known that in terms of stable splittings, one does not lose any information by this process. There are two advantages to this approach. One is that a result of Henn [8] implies that the ring of induced self-maps on the Fp-homology of BG is finite. Two is that one can now give a more explicit description of all the induced self-maps on Fp-homology for a large class of compact Lie groups which we call p-Roquette. The latter is exactly the class of compact Lie groups for which an appropriate density theorem is valid for a generalized Segal’s Burnside ring conjecture for compact Lie groups as shown by Minami [16] whose result built upon work of Feshbach on the original Segal’s Burnside ring conjecture for compact Lie groups [7]. In particular, every compact Lie group is p-Roquette if p is odd. With these reductions, one can follow a similar procedure for splitting BG∧ p as in the case when G is finite. Recall that a compact Lie group Q is said to be p-toral if it is an extension of a torus by a finite p-group. The main result of this paper is that when G is p-Roquette, one can reduce the stable splitting of BG∧ p to the study

Stable splittings of BP for some P of order thirty-two

In this paper, we use the method of Benson and Feshbach outlined in “Stable splittings of classifying spaces of finite groups” (in Topology, Vol. 31, No. 1 (1992)) to give the complete, 2-complete stable splitting of the classifying space BP for all but two of the groups P of order thirty-two which are not direct products in a nontrivial way. One of these two is Z 32 , and B( Z 32 ) is known to be indecomposable. For the other, P is the semidirect product of ( Z 2 ) 4 and solZ 2 . This is the only nonabelian group of order 32 which is not a direct product and has a subgroup isomorphic to ( Z 2 ) 4 . We also give some results which refine the method of Benson and Feshbach. We have found that BP is indecomposable for three of these groups, and these are the first examples of indecomposable BP for which P is a nonabelian 2-group. Poincaré series are given for each classifying space using Rusin's paper “The cohomology of the groups of order 32” (Mathematics of Computation, Vol. 53, No. 187 (1989)).

Stably splitting BG

In the early nineteen eighties, Gunnar Carlsson proved the Segal conjecture on the stable cohomotopy of the classifying space B G BG of a finite group G G . This led to an algebraic description of the ring of stable self-maps of B G BG as a suitable completion of the “double Burnside ring”. The problem of understanding the primitive idempotent decompositions of the identity in this ring is equivalent to understanding the stable splittings of B G BG into indecomposable spectra. This paper is a survey of the developments of the last ten to fifteen years in this subject.