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**A geometric interpretation of Lannes’ functor T.**
*(English)*
Zbl 0723.55006

Théorie de l’homotopie, Colloq. CNRS-NSF-SMF, Luminy/Fr. 1988, Astérisque 191, 87-95 (1990).

[For the entire collection see Zbl 0721.00021.]

From the authors’ introduction: “In this note we are concerned with a question raised by Lannes. In what follows R will denote a finite field of the form \({\mathbb{Z}}/p{\mathbb{Z}}\), homology and cohomology are always taken with coefficient in R and denoted by \(H_*X\) etc. For a space X let \(\{R_ sX\}_ s\) denote the Bousfield-Kan localization tower. We denote by \(B\tau\) the classifying space of the underlying Abelian group of R. Let \(P_ sX\) denote the s-Postnikov section of X. By a “space” we mean a Kan complex or a C.W. complex.

1.1. Theorem: If \(H^ iX<\infty\) for all \(i\geq 0\), then \(TH^*X\cong \varinjlim_ s H^*(P_ sR_ sX)^{B\tau}\), where T is Lannes’ functor. If, in addition, X is nilpotent then \(TH^*X\cong \varinjlim H^*(P_ sX)^{B\tau}\cong \varinjlim H^*(P_ sR_{\infty}X)^{B\tau}.\)

The proof of this theorem yields a new proof for the Lannes theorem that essentially asserts 1.1 for dimension zero and was the motivation for his question. The proof of the theorem is based on the following technical proposition:

1.2 Proposition: Let \(G\to E\to B\) be a principal fibration where G is a (topological or simplicial) group. Assume that in each dimension the R- cohomology of the mapping spaces \(E^{B_ t}\) and \(B^{B_ t}\) is finite. Then if the relation \(TH^*W\cong H^*W^{B_ t}\) is satisfied by \(W=E\) and \(W=B\) then it is also satisfied by \(W=G\).”

Remark: The finiteness assumption, noted by the referee, is necessary in order to use cohomological Eilenberg-Moore spectral sequence.”

From the authors’ introduction: “In this note we are concerned with a question raised by Lannes. In what follows R will denote a finite field of the form \({\mathbb{Z}}/p{\mathbb{Z}}\), homology and cohomology are always taken with coefficient in R and denoted by \(H_*X\) etc. For a space X let \(\{R_ sX\}_ s\) denote the Bousfield-Kan localization tower. We denote by \(B\tau\) the classifying space of the underlying Abelian group of R. Let \(P_ sX\) denote the s-Postnikov section of X. By a “space” we mean a Kan complex or a C.W. complex.

1.1. Theorem: If \(H^ iX<\infty\) for all \(i\geq 0\), then \(TH^*X\cong \varinjlim_ s H^*(P_ sR_ sX)^{B\tau}\), where T is Lannes’ functor. If, in addition, X is nilpotent then \(TH^*X\cong \varinjlim H^*(P_ sX)^{B\tau}\cong \varinjlim H^*(P_ sR_{\infty}X)^{B\tau}.\)

The proof of this theorem yields a new proof for the Lannes theorem that essentially asserts 1.1 for dimension zero and was the motivation for his question. The proof of the theorem is based on the following technical proposition:

1.2 Proposition: Let \(G\to E\to B\) be a principal fibration where G is a (topological or simplicial) group. Assume that in each dimension the R- cohomology of the mapping spaces \(E^{B_ t}\) and \(B^{B_ t}\) is finite. Then if the relation \(TH^*W\cong H^*W^{B_ t}\) is satisfied by \(W=E\) and \(W=B\) then it is also satisfied by \(W=G\).”

Remark: The finiteness assumption, noted by the referee, is necessary in order to use cohomological Eilenberg-Moore spectral sequence.”

Reviewer: M.Mimura (Okayama)

### MSC:

55Q05 | Homotopy groups, general; sets of homotopy classes |

55P60 | Localization and completion in homotopy theory |