An Application of Homology with Integer Coefficients Winding Number

1 An Application of Homology with Integer Coefficients Winding Number 

The winding number of a closed curve, and more generally, the degree of a continuous map between spheres of the same dimension, are invariants, which are useful in many situations. We shall need these to define the so-called incidence numbers, which in turn allow one to give an explicit description of cellular homology. 

The main reason why CW complexes are so handy for concrete computations of homology groups is that it turns out that one can substitute the simplicial chain complex with another, usually much smaller, chain complex, which we now proceed to define. 

Together, equations (3.18) and (3.19) define the cellular chain complex oi a given CW complex  

Although in general it may be hard to compute the incidence numbers, in particular situations it is often geometrically clear what they are. 

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