A relative end theorem for stratified spaces
by
Frank Quinn
Virginia Tech

Suppose f:X --> Y is a map with an end in the sense of (F. Quinn, Ends of maps II, Invent. Math. 68 (1982) 353-424). If X is a manifold and f is appropriately tame then obstructions to completing f are described in this reference. This result is relative in the sense that if X is a manifold with boundary, and the boundary has a completion then the completion on X can be chosen to extend the given one. Here we describe a version in which X is stratified rather than a manifold, and a completion is given on the lower strata.

Specifically suppose X contains X₀, Y contains Y₀, and f is a strict map of pairs (preserves subspaces and complements). Suppose there is an isomorphism X₀ =~ B×(0, \infty) so that the composition B×(0, \infty) --> Y₀ extends to the half-open interval B×(0, \infty]. Finally suppose that X-X₀ is a manifold (of dimension greater than 5). The objective is to find an open collar structure on the end of X that extends the one given on X₀, and so that the composition to Y extends to the half-open collar. There is an obstruction to this, of the usual spectral cosheaf homology type. Instead of giving this directly we construct a non-relative end, and assert that the original relative problem has a solution if and only if the associated non-relative version has a solution. The older theory then gives an obstruction for the associated version. This obstruction-free formulation is itself quite useful.

Consider Y \cup X₀ with the union with the topology induced by f as a stratified space with top stratum X-X₀ and singular set Y \cup X₀.

Theorem. Under appropriate tameness hypotheses (eg\. X \cup Y is a homotopically stratified set) X has a completion over Y extending the one given on X₀ if and only if X-X₀ has a completion over Y \cup X₀.

This result was inspired by a partial version in the Connolly-Vajiac paper on ends of stratified spaces (Invent. Math, to appear). It can be used to describe obstructions to mapping cylinder neighborhoods for pure subsets of a stratified space.

Date received: March 1, 1998