On the homotopy groups of the self-equivalences of linear spheres

Abstract

Let S(V ) be a complex linear sphere of a finite group G. Let S(V ) * n denote the n-fold join of S(V ) with itself and let aut G (S(V ) * ) denote the space of G-equivariant self-homotopy equivalences of S(V ) * n . We show that for any k 1 there exists M > 0 that depends only on V such that |π k aut G (S(V ) * n )| M for all n 0.

Stabilization of the homotopy groups of the self equivalences of linear spheres

“…Klaus proved that for any k ≥ 1 the groups π k map G (S(U ≤n ), S(U ≤n )), where id is the basepoint, are finite for all sufficiently large n [5, Proposition 2.5]. The author improved this result in [7], giving bounds for their order (uniform in n for each k). However, to establish Theorem 1.1 in its full generality spectral sequences become unmanageable.…”