Subgroups of the group of self-homotopy equivalences

Details Subgroups of the group of self-homotopy equivalences Subgroups of the group of self-homotopy equivalences

2000-10-12

arxiv.org/abs/math/0010121v1

Contemp. Math., Vol. 274 (2001), 21--32

Mathematics

Algebraic Topology

To appear, in Contemp. Math

Scientific paper

Denote by E(Y) the group of homotopy classes of self-homotopy equivalences of a finite-dimensional complex Y. We give a selection of results about certain subgroups of E(Y). We establish a connection between the Gottlieb groups of Y and the subgroup of E(Y) consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of Y, denoted by E_#(Y). We give an upper bound for the solvability class of E_#(Y) in terms of a cone decomposition of Y. We dualize the latter result to obtain an upper bound for the solvability class of the subgroup of E(Y) consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups with various coefficients. We also show that with integer coefficients, the latter group is nilpotent.

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