Homotopy fixed-point methods for Lie groups and finite loop spaces.

*(English)*Zbl 0801.55007A loop space is a space \(X\) together with another connected space \(BX\) and a homotopy equivalence \(e: X \to \Omega BX\). The loop space \(X\) is said to be finite if \(H_ *(X;\mathbb{Z}) = \bigoplus_ i H_ i(X;\mathbb{Z})\) is a finitely generated abelian group. The main theorem of the present paper says that, if \(X\) is a finite loop space and \(p\) is a prime number, the algebra \(H^*(BX;\mathbb{F}_ p)\) is finitely generated. A compact Lie group \(G\) is a special case of a finite loop space; it is well-known that the algebra \(H^*(BG;\mathbb{F}_ p)\) is finitely generated.

The most interesting aspect of the authors’ work is the development of an analogue of the theory of compact Lie groups in the realm of homotopy theory. They define a \(p\)-compact group as a loop space \(X\) with the following three properties:

a) \(H^*(X;\mathbb{F}_ p)\) is of finite dimension over \(\mathbb{F}_ p\).

b) \(\pi_ 0(X)\) is a finite \(p\)-group.

c) For \(i \geq 1\), \(\pi_ i(X)\) is a finitely generated module over the \(p\)-adic integers \(\mathbb{Z}_ p\).

It is remarkable that \(p\)-compact groups behave very much like compact Lie groups. In particular, with the appropriate definitions, there are maximal tori and Weyl groups. Roughly speaking, many notions of Lie group theory can be formulated in terms of fixed point sets of certain group operations; the homotopy theoretical generalization is then obtained by considering the corresponding homotopy fixed point sets. The proofs rely on \(\mathbb{F}_ p\)-completions and on Lanne’s theory.

The most interesting aspect of the authors’ work is the development of an analogue of the theory of compact Lie groups in the realm of homotopy theory. They define a \(p\)-compact group as a loop space \(X\) with the following three properties:

a) \(H^*(X;\mathbb{F}_ p)\) is of finite dimension over \(\mathbb{F}_ p\).

b) \(\pi_ 0(X)\) is a finite \(p\)-group.

c) For \(i \geq 1\), \(\pi_ i(X)\) is a finitely generated module over the \(p\)-adic integers \(\mathbb{Z}_ p\).

It is remarkable that \(p\)-compact groups behave very much like compact Lie groups. In particular, with the appropriate definitions, there are maximal tori and Weyl groups. Roughly speaking, many notions of Lie group theory can be formulated in terms of fixed point sets of certain group operations; the homotopy theoretical generalization is then obtained by considering the corresponding homotopy fixed point sets. The proofs rely on \(\mathbb{F}_ p\)-completions and on Lanne’s theory.

Reviewer: W.Singhof (Düsseldorf)

##### MSC:

55P35 | Loop spaces |

57T99 | Homology and homotopy of topological groups and related structures |

57S99 | Topological transformation groups |