Hatcher, *Algebraic Topology*, Chapter 2, Section 1

**5. Compute the simplicial homology groups of the Klein bottle using the -complex structure described at the beginning of the section.**

*Proof.* Recall that the -complex structure referred to here is as follows:

Under this structure, we have that , ,^{1} and . This gives rise to the following chain complex where :

The maps behave as follows: and . In particular, then, , which implies that , i.e. . Moreover, the (reduced) image of can be computed by way of the following matrix computation:

.

Hence, . Fortunately, the other maps are easier to analyze: so that and , and implies that , . Now then:

,

,

and

.

Of course, the above representation of is hardly satisfying, so to remedy that, we turn to a little algebra.

In the quotient, . Note that implies that the homology group will have torsion in the form of a summand. Otherwise, precisely when . Therefore, the “numerator group” is equivalent to the group , while the “denominator” group is precisely . Hence, has the form

.

1. The choice of

for the second component is allowable due to the fact that the matrix with rows

row reduces to the identity matrix, i.e. the basis

is equivalent to the basis

. Moreover, considering the computations that follow, this basis makes the final computation more clear. Even clearer would have been the (also equivalent) basis

, a fact I didn’t realize until later.

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