Hurewicz map

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Explicit definition

Let X be a path-connected space. For n a positive integer, the n^{th} Hurewicz map based at x_0 of X is a 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 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.

To note that this gives a well-defined map on \pi_n(X,x_0), 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. Further information: Hurewicz map is well-defined

Hands-off definition

Here is an alternative description of the map. We use the fact that f: S^n \to X induces a map between H_n(S^n) and H_n(X). But H_n(S^n) = \mathbb{Z} and we can thus simply look at the image of the generator of this, to give an element in H_n(X).


The image of the Hurewicz map

The image of the Hurewicz map is a subgroup comprising those singular homology classes that are represented by a singular simplex with the property that all points of the boundary get mapped to the basepoint x_0.

In particular, the Hurewicz map being surjective means that every continuous map from \Delta^n to X (and in fact, every formal sum of such continuous maps) is homologous to a continuous map with the property that the entire boundary \partial \Delta^n is mapped to x_0.

The kernel of the Hurewicz map

The kernel of the Hurewicz map comprises those homotopy classes of maps from S^n to (X,x_0) that are nullhomologous. In the case n = 1, the explanation lies in non-commutativity, i.e., by cutting and rearranging the pieces of the map, we can get a nullhomotopic map.

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