- #1
Bacle
- 662
- 1
Hi, everyone:
Given a smooth, orientable manifold X, we turn Aut(X)
the collection of all self-diffeos. of X into
a topological space, by using the compact-open
topology. Aut(X) is also a group under composition.
The mapping class group M(X) of X is defined as the
quotient:
M(X):= Aut(X)/Aut_id(X)
where Aut_id(X) is the path-component of the
identity map --which coincides with the isotopy
class of IdX in the compact-open topology.
(group operation is composition, of course)
My question:
In order for M(X) to be a group, we must have
Aut_id(X) be a normal subgroup of X. How do we know
that Aut_id(X) is normal in X?. I think we need for
X to be a topological group or something, but I
am not sure.
Thanks For any Help.
Given a smooth, orientable manifold X, we turn Aut(X)
the collection of all self-diffeos. of X into
a topological space, by using the compact-open
topology. Aut(X) is also a group under composition.
The mapping class group M(X) of X is defined as the
quotient:
M(X):= Aut(X)/Aut_id(X)
where Aut_id(X) is the path-component of the
identity map --which coincides with the isotopy
class of IdX in the compact-open topology.
(group operation is composition, of course)
My question:
In order for M(X) to be a group, we must have
Aut_id(X) be a normal subgroup of X. How do we know
that Aut_id(X) is normal in X?. I think we need for
X to be a topological group or something, but I
am not sure.
Thanks For any Help.