Vecteurs de Witt non commutatifs et représentabilité de l’homologie modulo \(p\). (Noncommutative Witt vectors and representability of homology modulo \(p\)). (French) Zbl 0769.55011
This paper deals with a space \(T_ 1(n)\) which has nice representability properties for homology classes. Let \(A\) denote the \(\bmod p\) Steenrod algebra, and let \(G(n)\) denote the free right unstable \(A\)-module on a class of degree \(n\). Then \(T_ 1(n)\) is the unique space, up to homotopy, such that (i) \(H_ *T_ 1(n) \approx \Sigma G(n)\), (ii) \(\Sigma\Omega T_ 1(n) \to T_ 1(n)\) induces a surjection in \(\bmod p\) homology, and (iii) \(H_ *(T_ 1(n)\); \(Z[1/p]) = 0\). Then \([T_ 1(n),\Sigma Y] \to H_ n(Y)\) is surjective for all spaces \(Y\), while \([T_ 1(n),Y] \to \text{Hom}_{{\mathcal H}}(H_ *\Omega T_ 1(n),H_ *(\Omega_ 0Y)\) is surjective for all spaces \(Y\), and is bijective if \(Y\) is an Eilenberg-MacLane space or of \(n \not\equiv\pm1\bmod 2p\). Here \(\mathcal H\) is the category of unstable \(A\)-Hopf algebras.
Alternatively, \([T_ 1(n),Y]\) maps to \(\text{Hom}_{{\mathcal H}^{F_ p}}(W(n),H_ *\Omega_ 0Y)\) surjectively, and bijectively if \(Y\) is an Eilenberg-MacLane space or if \(n \not\equiv \pm 1\bmod 2p\). Here \(W(n)\) is the projective cover in \(\mathcal H\) of the tensor algebra on a generator of degree \(n\), and \({\mathcal H}^{F_ p}\) is the category of graded cocommutative connected Hopf algebras over \(F_ p\). The Witt vectors in the title are related to \(W(n)\).
Alternatively, \([T_ 1(n),Y]\) maps to \(\text{Hom}_{{\mathcal H}^{F_ p}}(W(n),H_ *\Omega_ 0Y)\) surjectively, and bijectively if \(Y\) is an Eilenberg-MacLane space or if \(n \not\equiv \pm 1\bmod 2p\). Here \(W(n)\) is the projective cover in \(\mathcal H\) of the tensor algebra on a generator of degree \(n\), and \({\mathcal H}^{F_ p}\) is the category of graded cocommutative connected Hopf algebras over \(F_ p\). The Witt vectors in the title are related to \(W(n)\).
Reviewer: D.Davis (Bethlehem)
MSC:
| 55S37 | Classification of mappings in algebraic topology |
| 55P35 | Loop spaces |
| 55P42 | Stable homotopy theory, spectra |
Keywords:
Brown-Gitler spectra; unstable Hopf-algebras over the \(\bmod p\) Steenrod algebra; representing homology classes; maps from a model space; \(\bmod p\) Steenrod algebra; projective cover; Witt vectorsReferences:
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