5

For the generalized Birman exact sequence, on A Primer on Mapping Class Groups, it's given by $$1\to \pi_1(C_n(S,n))\to \text{MCG}(S, P)\to \text{MCG}(S)\to 1,$$ where the group $\text{MCG}(S, P)$ is the group of isotopy classes of homeomorphisms of $S$ fixing $P$ as a set.

In the book, it says it's valid when $\chi(S)<0$ or $S$ is a disk $D^2$. But it seems that $\text{MCG}(D^2, P)\not=\text{MCG}(D^2\setminus \{P\})$. For example when $P$ is a single point, then $D^2\setminus \{P\}$ is an annulus, thus $\text{MCG}(D^2\setminus \{P\})=\mathbb Z$, but by generalized Birman exact sequence $\text{MCG}(D^2, P)=1$

The usual Birman exact sequence (Theorem 4.6) doesn't include $D^2$, but the generalized Birman exact sequence does include $D^2$, so I get $\text{MCG}(D^2, P)=B_1=1$; I am not sure if I missed anything, maybe the generalized Birman exact sequence doesn't include the case of $D^2$ for $|P|=1$ (so that it agrees with the usual Birman exact sequence)? Section 1.1.1 of the book also mentions that "Marked points and punctures carry the same topological information", so maybe $\text{MCG}(D^2, P)=\text{MCG}(D^2\setminus \{P\})$?

If $\text{MCG}(D^2, P)\not=\text{MCG}(D^2\setminus \{P\})$, I am wondering when $\text{MCG}(S, P)=\text{MCG}(S\setminus \{P\})$? Is that only true when $S$ is hyperbolic? I can't find a reference.

user1249906
  • 109

1 Answers1

3

For a compact annulus $A$, meaning that $A$ is homeomorphic to $S^1 \times [0,1]$, it is true that $\text{MCG}(A) \approx \mathbb Z$.

But $D^2 \setminus \{P\}$ is not a compact annulus. Instead, it is homeomorphic instead to $S^1 \times [0,1)$. And so yes, $\text{MCG}(D^2 \setminus \{P\})$ is a trivial group.

Your closing question is over budget for your post; see our policy on one question per post. But I'll address that question very briefly. The isomorphism between $\text{MCG}(S,P)$ and $\text{MCG}(S \setminus \{P\})$ holds for all surfaces. A good, conceptual, topological proof of this statement can be obtained using the concept of ends of the topological space $S \setminus \{P\}$.

Lee Mosher
  • 138,765