Paths and cylinders
The first nonzero homotopy group and ordinary/singular homology group of a simply-connected topological space occur in the same dimension and are isomorphic.
For a pointed topological space, the Hurewicz homomorphism is the function
from the th homotopy group of to the th singular homology group defined by sending
a representative singular -sphere in to the push-forward along of the fundamental class .
A proof is spelled out for instance with theorem 2.1 in (Hutchings).
This appears for instance as theorem 4.32 in
Lecture notes include
Revised on November 17, 2013 00:54:38
by Urs Schreiber