Vipul: Created page with '==Statement== ===Loose statement=== Let $X$ be a path-connected space. For $n$ a positive integer, we want to show that the $n^{th}$ [[Hure...'

2011-01-09T21:59:40Z

Created page with '==Statement== ===Loose statement=== Let <math>X</math> be a path-connected space. For <math>n</math> a positive integer, we want to show that the <math>n^{th}</math> [[Hure...'

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==Statement==

===Loose statement===

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

<math>\pi_n(X,x_0) \to H_n(X)</math>

where <math>\pi_n(X,x_0)</math> is the <math>n^{th}</math> [[homotopy group]], and <math>H_n(X)</math> is the <math>n^{th}</math> [[singular homology group]].

===The strict map===

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

<math>\eta:\Delta^n \to S^n</math>

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

Now given any based continuous map <math>f: (S^n,*) \to (X,x_0)</math>, consider <math>f \circ \eta</math>. This gives a <math>n</math>-[[singular chain]] in <math>X</math>, 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 <math>f_1</math> and <math>f_2</math> are [[homotopic map]]s as based continuous maps from <math>(S^n,*)</math> to <math>(X,x_0)</math>, then <math>f_1 \circ \eta</math> and <math>f_2 \circ \eta</math> are both in the same homology class.

==Proof==

{{fillin}}

Hurewicz map is well-defined - Revision history

Vipul: Created page with '==Statement== ===Loose statement=== Let X be a path-connected space. For n a positive integer, we want to show that the n^{th} [[Hure...'

Vipul: Created page with '==Statement== ===Loose statement=== Let $X$ be a path-connected space. For $n$ a positive integer, we want to show that the $n^{th}$ [[Hure...'