Stable splittings of mapping spaces

Bödigheimer, Carl-Friedrich

doi:10.1007/bfb0078741

Cited by 49 publications

(82 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The construction and proof of the second equivalence carry over from the proof of the corresponding statement in [1] Corollary 3. Let M be a parallelizable, compact, m-dimensional manifold with boundary.…”

Section: Lemmamentioning

confidence: 99%

“…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, S m Z), which stably splits. More precisely, there is a weak equivalence:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A generalization of Snaith-type filtration

Arone

1999

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. 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. IntroductionIt 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, S m Z), which stably splits. More precisely, there is a weak equivalence:where C(M, ∂M ; n) stands for the space of n-tuples of distinct points in M , where all n-tuples whose intersection with ∂M 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 Ω m Σ m X. This later splitting is sometimes refered to as the Snaith splitting (at least in the case m = ∞), 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 → 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 = S m Z. 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 factorsReceived by the editors July 21, 1994 and, in revised form, February 4, 1997. 1991 Mathematics Subject Classification. Primary 55P99. are homogeneous functors (of Z). Therefore, the functorsare excisive of degree n. Moreover, it is, in fact, true that the (weak) mapis (n + 1)k-connected if the space Z is k-connected. Therefore, by universality, the right hand side of (0.1) is nothing but the Taylor tower of the functor Z → QMap(M, S m Z), where Z ∈ T * . One can think of it as the "Maclaurin tower" of QMap(M, S m Z). Hence the generalization that we seek in this paper is the one of identifying the Taylor tower of the functor QMap * (K, Y ), where K is a finite CWcomplex and Y ∈ T X . More concretely, we want to answer the following question: given a space Y containing X as a retract, how can we describe QMap * (K, Y ) as a sequence of extensions of QMap * (K, X) by homogeneous functors? Let us discuss the case X = * (Maclaurin tower) first. We use implicitly the fact that the differentials of ...

show abstract

Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A generalization of Snaith-type filtration

Arone

1999

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Let D n ⊂ R n be the unit disc. The sequence [2]). Now Σ n X ⊂ C(S n , D n ; X) is a strong deformation retract, seen as the subspace of configurations with a single labelled point [7].…”

Section: Configuration Spacesmentioning

confidence: 99%

“…There is a geometric version of this corollary. Recall that C(M, X) is stably homotopy equivalent to the wedge sum of D k (M, X), indexed over k [2]. The Browder map defines a stable map b k :…”

Section: Definition 2 the Browder Action Is The Homomorphismmentioning

confidence: 99%

Configuration spaces on the sphere and higher loop spaces

Salvatore

2004

Math. Z.

View full text Add to dashboard Cite

We show that the homology over a field of the space Λ n Σ n X of free maps from the n-sphere to the n-fold suspension of X depends only on the cohomology algebra of X and compute it explicitly. We compute also the homology of the closely related labelled configuration space C(S n , X) on the n-sphere with labels in X and of its completion, that depends only on the homology of X. In many but not all cases the homology of C(S n , X) coincides with the homology of Λ n Σ n X. In particular we obtain the homology of the unordered configuration spaces on a sphere.MSC (2000): 55P48, 55R80, 55S12.

show abstract

“…In 1978, F. Cohen and L. Taylor [5] were able to compute the cohomology of the configuration spaces for a certain class of manifolds, and applied these results to describe the Gelfand-Fuks cohomology of these manifolds. Also, there has been interest in the configuration spaces as they appear as approximations to certain function spaces (see [2,9]). In this paper we study the configuration spaces and the function spaces as outlined by Bott-Segal.…”

Section: Introductionmentioning

confidence: 99%

The cohomology of certain function spaces

Bendersky¹,

Gitler²

1991

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We study a spectral sequence converging to the cohomology of the configuration space of n ordered points in a manifold. A chain complex is constructed with homology equal to the E2 term. If the field is the rationals and the manifold is formal then the spectral sequence is shown to collapse. The results are applied to compute the Anderson spectral sequence converging to the cohomology of a function space. In 1969-1970, Gelfand and Fuks [6] wrote a series of papers on the continuous cohomology of the Lie algebra of vector fields on a manifold M. They introduced a pair of spectral sequences for such a study. Based on the results of Gelfand-Fuks, Bott and Fuks conjectured that the continuous cohomology of the Lie algebra of vector fields was the usual cohomology of a certain function space, namely, that of the space of sections of a certain associated bundle to the tangent bundle of M.Haëfliger [7] and Bott and Segal [3] in 1978, using entirely different methods, proved the Bott-Fuks conjecture. The method followed by Bott and Segal was to utilize the ideas of Anderson.The cohomology of "configuration spaces" appears in the computation of the Tij-term of the above spectral sequences. In 1978, F. Cohen and L. Taylor [5] were able to compute the cohomology of the configuration spaces for a certain class of manifolds, and applied these results to describe the Gelfand-Fuks cohomology of these manifolds. Also, there has been interest in the configuration spaces as they appear as approximations to certain function spaces (see [2,9]). In this paper we study the configuration spaces and the function spaces as outlined by Bott-Segal. In both instances the cohomology can be obtained from a

show abstract

Stable splittings of mapping spaces

Cited by 49 publications

References 21 publications

A generalization of Snaith-type filtration

A generalization of Snaith-type filtration

Configuration spaces on the sphere and higher loop spaces

The cohomology of certain function spaces

Contact Info

Product

Resources

About