# Algebraic Topology

• February 1st 2011, 09:28 AM
Turloughmack
Algebraic Topology
If h is a path in X starting at x0 and ending at x1 then we
have deﬁned an isomorphism βh : π1 (X, x1 ) → π1 (X, x0 ).
Now suppose that x0 = x1 .
In that case βh : π1 (X, x0 ) → π1 (X, x0 ).
In other words βh is an automorphism of π1 (X, x0 ). Can you describe this
automorphism more concretely?
• February 1st 2011, 06:49 PM
tonio
Quote:

Originally Posted by Turloughmack
If h is a path in X starting at x0 and ending at x1 then we
have deﬁned an isomorphism βh : π1 (X, x1 ) → π1 (X, x0 ).
Now suppose that x0 = x1 .
In that case βh : π1 (X, x0 ) → π1 (X, x0 ).
In other words βh is an automorphism of π1 (X, x0 ). Can you describe this
automorphism more concretely?

It is exactly the same as the original isomorphism $\beta$ which, I presume, is an inner one.

Tonio