BP cohomology of mapping spaces from the classifying space of a torus to some p torsion free space

pefav - Jean Lannes

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7 pages

English

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BP-cohomology of mapping spaces from the classifying space of a torus to some p-torsion free space 1. Introduction Let p be a fixed prime number, V a group isomorphic to (Z/p)d for some integer d and BV its classifying space. The exceptional properties of the mod p cohomology of BV , as an unstable module and algebra over the Steenrod algebra, have led to the calculation of the mod p cohomology of the mapping spaces with source BV as the image by a functor TV of the mod p cohomology of the target ([La2]). This determination is linked to the solution of the Sullivan conjecture concerning the space of homotopy fixed points for some action of a finite p-group ([Mi]). It is also an essential component of the homotopy theory of Lie groups initiated by Dwyer et Wilkerson ([DW]). We will see that we can deduce from the theory of the T functor (and from its equivariant version) a similar theory relative to the BP-cohomology of the mapping spaces with source the classifying space of a torus when the target is a space whose cohomology with p-adic coefficients is torsion free (p-torsion free space). We start with recalling the theory of the TV functor. Let K be the category of unstable algebras over the Steenrod algebra (the mod p cohomology of a space X, which we denote by H?X, is a typical object of K).

steenrod algebra

bp?-module

ring spectrum

unstable algebra

space hom

h?x ?

mapping spaces

Sujets

Lannes

Steenrod algebra

Spectrum of a ring

maths

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Publié par	pefav
Nombre de lectures	6
Langue	English

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BP-cohomology of mapping spaces from the classifying space of a torus to somep-torsion free space

Introduction

d Letpbe a ﬁxed prime number,Va group isomorphic to (Z/psome integer) for dand BV its classifying space. The exceptional properties of the modpcohomology of BV, as an unstable module and algebra over the Steenrod algebra, have led to the calculation of the modpcohomology of the mapping spaces with source BVas the image by a functor TVof the modpcohomology of the target ([La2]). This determination is linked to the solution of the Sullivan conjecture concerning the space of homotopy ﬁxed points for some action of a ﬁnitep-group ([Miis also an essential]). It component of the homotopy theory of Lie groups initiated by Dwyer et Wilkerson ([DW]). We will see that we can deduce from the theory of the T functor (and from its equivariant version) a similar theory relative to the BP-cohomology of the mapping spaces with source the classifying space of a torus when the target is a space whose cohomology withp-adic coeﬃcients is torsion free (p-torsion free space). We start with recalling the theory of the TVfunctor. LetKbe the category of unstable algebras over the Steenrod algebra (the modpcohomology of ∗ a spaceX, which we denote by HX, is a typical object ofK). It is a subcategory of the abelian categoryEof gradedFpJ. Lannes deﬁnes the functor T-vector spaces. Vas the left adjoint of the ∗ ∗ functorK → K,N7→H BV⊗N. The Bexceptional properties of the unstable algebra H Vcome down to the following statement:

0 00 Proposition1.1.LetM→MandM→Mbe formal linear combinations of morphisms 0 00 ofKsuch that the induced sequenceM→M→MofEis exact; then so is the sequence 0 00 TVM→TVM→TVM. We will say that the functor TVonKis exact. LetXbe a space (an object of the categorySof simplicial sets, ﬁbrant in what follows). We lethom(BV, X) denote the space of maps from BVtoXcounit of the adjunction in. The S, ∗ ∗ ∗ BV×hom(BV, X)→Xinduces a morphism HX→H BV⊗Hhom(BV, X) thus a morphism ∗ ∗ TVHX→Hhom(BV, Xproperties of T). The Vare so that this last morphism is very often an isomorphism ([La2]); we have for example:

Theorem1.2 ([La2], [DS]).LetXbe a ﬁbrant degree wise ﬁnite simplicial set, having for every choice of the base point a ﬁnite number of non trivial homotopy groups, each of them being a ﬁnite p-group (we say thatXis a ﬁnitep-space); then the natural morphism

∗ ∗ TVHX→Hhom(BV, X)

is an isomorphism. ∗ ∗ ∗ We note that if HXis degree wise ﬁnite dimensional but not TVHXthen TVHXand ∗ Hhom(BV, XNevertheless, theorem 1.2 can be generalized) do not have the same cardinal. to ﬁltered limits of ﬁnitep-spaces (pro-p-spaces) and their limit cohomology. Thus replacing the b ∗ spaceXby its pro-p-completionX(−), one interprets TVHXas the limit modpcohomology of b the pro-p-spacehom(BV, X(−)) ([Mowill do it implicitly in what follows.]). We We have an equivariant version of the TV-functor theory ([La2]): Suppose thatXis given with hV some action ofVdenote by. We Xthe space ofV-equivariant maps from the universal covering EVof BVtoX, which we call the homotopy ﬁxed points space ofXunder the action ofV. Let

A part of the results stated below corresponds to joint work with Jean Lannes and has led to a common preprint ([DL]).