Topology + Algebra and Analysis
32nd Summer Conference on Topology and Its Applications
Let λ be a limit ordinal and consider a directed system of topological groups (Gα)α < λ with topological embeddings as bonding maps and its directed union G=∪α < λGα. There are two natural topologies on G: one that makes G the direct limit (colimit) in the category of topological spaces and one which makes G the direct limit (colimit) in the category of topological groups.
For λ = ω it is known that these topologies almost never coincide (Yamasaki's Theorem).
In my talk last year, I introduced the Long Direct Limit Conjecture, stating that for λ = ω1 the two topologies always coincide.
This year, I will introduce one particular example of such a direct limit: The groups of compactly supported homeomorphisms of the Long Line which is naturally such a directed union of topological groups. I will explain why on this group the two direct limit topologies mentioned above agree (and are equal to the compact open topology). Unfortunately this method only works in dimension one and breaks down as soon as one wants to consider groups of homeomorphisms of the Long Plane or similar two dimensional manifolds.
Copyright © 2017, the Authors
Dahmen, Rafael and Lukács, Gábor, "Compactly Supported Homeomorphisms as Long Direct Limits" (2017). Summer Conference on Topology and Its Applications. 61.