Hurewicz map is well-defined

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Statement

Loose statement

Let X be a path-connected space. For n a positive integer, we want to show that the n^{th} Hurewicz map based at x_0 of X is a well-defined map:

\pi_n(X,x_0) \to H_n(X)

where \pi_n(X,x_0) is the n^{th} homotopy group, and H_n(X) is the n^{th} singular homology group.

The strict map

The map is defined as follows. First define a map:

\eta:\Delta^n \to S^n

which essentially uses the identification of S^n with the quotient of \Delta^n by the collapse of its boundary to a single point, i.e., a homeomorphism \Delta^n/\partial \Delta^n \to S^n.

Now given any based continuous map f: (S^n,*) \to (X,x_0), consider f \circ \eta. This gives a n-singular chain in X, and its homology class is precisely the element we are looking for.

What we need to show

To note that this is indeed well-defined, we need to show that if f_1 and f_2 are homotopic maps as based continuous maps from (S^n,*) to (X,x_0), then f_1 \circ \eta and f_2 \circ \eta are both in the same homology class.

Proof

Fill this in later