BP cohomology of mapping spaces from the classifying space of a torus to some p torsion free space
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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).
Voir
- steenrod algebra
- bp?-module
- ring spectrum
- unstable algebra
- space hom
- h?x ?
- mapping spaces
- tv
- hs?
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English
BP-cohomology of mapping spaces from the classifying space of a torus to somep-torsion free space
1.
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]).
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