Xiaolei Wu - Compactly-supported classes in the homology of big mapping class groups

Abstract

I will start the talk with an overview about the theory of big mapping class groups. Then I will discuss some recent advances on the homology of big mapping class groups. In particular, I want to talk about some of recent work with Martin Palmer on when a homology class of the big mapping class groups could have compact support.

Content

We will be studying mappings class groups of infinite surfaces. Some example of infinite type surfaces are

• Loch Ness monster surface
• Jacobs Ladder surface
• etc.

The Big Mapping class group is just a mapping class group of a surface of infinite type. Big mapping class groups are usually infinitely generated.

Compactly supported mapping class group

We look at the group which is compactly supported $Mo{d}_c$, i.e. it only changes the space on a compact set.

Theorems Let $L$ be Lochnessmonster $H^{\ast }(Mo{d}_c(L),\mathbb{Q})\cong \mathbb{Q}[...]$ We can study the mapping class group by looking at its homology group (Palmer-W.2022)