Ken-ichi Maruyama

Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups

Abstract For a based, 1-connected, finite CW-complex X, we study the following subgroups of the group of homotopy classes of self-homotopy equivalences of X: e ∗ (X) , the subgroup of homotopy classes which induce the identity on homology groups, e ∗ (X) , the subgroup of homotopy classes which induce the identity on cohomology groups and e#dim + r(X), the subgroup of homotopy classes which induce the identity on homotopy groups in dimensions ⩽ dim X + r. We investigate these groups when X is a Moore space and when X is a co-Moore space. We give the structure of the groups in these cases and provide examples of spaces for which the groups differ. We also consider conditions on X such that e ∗ (X) = e ∗ (X) and obtain a class of spaces (including compact, oriented manifolds and H-spaces) for which this holds. Finally, we examine e#dim + r(X) for certain spaces X and completely determine the group when X = Sm × Sn and X = CPn ∨ S2n.

NILPOTENT SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES

In this paper, we study subgroups of self-homotopy equivalences associated to generalized homology theories. We generalize Dror-Zabrodsky’s nilpotency theorem on the group of self-homotopy equivalences.

_{*}-kernels of Lie groups

We study a filtration on the group of homotopy classes of self maps of a compact Lie group associated with homotopy groups. We determine these filtrations of SU(3) and Sp(2) completely. We introduce two natural invariants lzp(X) and sz p (X) defined by the filtration, where p is a prime number, and compute the invariants for simple Lie groups in the cases where Lie groups are p-regular or quasi p-regular. We apply our results to the groups of self homotopy equivalences.

Finiteness properties of self-equivalence groups of rationalco-H-spaces

The group of self-homotopy equivalences of a finite complex which is rationally aco-H-space is studied. Some finiteness properties are obtained. Two subgroups consisting of elements which induce the identity on homotopy or homology groups are also studied. Examples are included showing these results are best possible.