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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.