By P. Hoffman, R. Piccinini, D. Sjerve

ISBN-10: 3540089306

ISBN-13: 9783540089308

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The further construction of the homology theory with coefficients in G does not differ from the case of integer coefficients.

The case (C) is more difficult, since there we cannot get rid of the new tangents. The solution is to endow the common tangents with signs ±1 in such a way that the tangents from each arising pair have distinct signs and hence cancel each other out under counting. Let l be a line which touches curves f and g at points f (t0 ) and g(τ0 ) respectively, where t and τ are the parameters on the two copies of S 1 . Assign to the point f (t0 ) a sign δ1 = ±1 depending on the direction of the turn (positive or negative) of the velocity vector v(t) = f (t) when the parameter t passes through the value t = t0 .

Transforming it to the canonic form (and removing superfluous rows and columns), we get the matrix 02 06 , which yields H1 (X) = Z2 ⊕ Z6 . a1 a3 a4 a5 a6 4 5 7 6 3 a4 a5 1 8 2 a1 a6 a3 1 1 1 1 _1 2 1 1 _1 2 2 1 Figure 27. Choosing generators and writing down a relation matrix of the first homology group. Exercise 55. Calculate the first homology group of the Klein bottle. Along with simplicial and cellular homologies, one can use homologies of other types, for instance, singular ones. The difference between the singular homology and the simplicial and cellular ones is in the method of assignment of a chain complex to a given space.

### Algebraic Topology by P. Hoffman, R. Piccinini, D. Sjerve

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