Massey Exercise 2.3.1


Let and let and be two path classes from to . We can characterize the conditions under which and give rise to the same isomorphism by observing the following.

Suppose and are the same. Then we have, for all :

Denote , and notice that and . Then we have . Since this holds for all , we have that .

By assuming that , we can reverse the argument above to conclude that .

So we have that the isomorphisms induced by and are identical if and only if .