New York Journal of Mathematics
Volume 24 (2018) 261-277


James West

Absolute retract involutions of Hilbert cubes: Fixed point sets of infinite codimension

view    print

Published: March 4, 2018
Keywords: Hilbert cube, involution, absolute retract
Subject: Primary: 57N20, 54C55; Secondary: 57S17, 54C15

Let α :Q→ Q be an involution of a Hilbert cube with fixed point set Qα that has Property Z in Q. The first main result of this paper is Theorem 3.1: Assume that (Q,α) is an absolute retract in the category of metric spaces with involutions and equivariant maps. If T⊆ Q is an equivariant retract of Q containing Qα that is an inequivariant Z-set in Q, then for any equivariant retraction r:Q→ T, Q is equivariantly homeomorphic with the mapping cylinder M(r;T) of r reduced at T. The second main result is part of Theorem 3.3: Qα is an equivariant strong deformation retract of Q if and only if Q is equivariantly homeomorphic with
Qα× Πi≧1Ii
equipped with the involution that reflects each interval coordinate Ii across its mid-point.

Author information

Department of Mathematics, Cornell University, Ithaca, NY 14053-4201, USA