algebraic-topologycw-complexes
It is easy to compute the Euler characteristic of the Mobius band and the Klein bottle using their CW decompositions. However I have an exercise which requires to use the formula $$\chi(X)=\chi(A)+\chi(B)-\chi(C),$$ where X is obtained by gluing two CW complexes $A$ and $B$ along their subcomplex $C$.
I have not any idea how to do that. Can someone help me?
Thanks a lot!
Let $M$ be a closed connected $n$ manifold then $H_i(M;\mathbb Z)=0$ for each $i>n$. If in addition $M$ is orientable then $H_n(M;\mathbb Z)\cong \mathbb Z$.
For a proof see Hatcher Theorem 3.26.
Apply this to $M=S^1$ which is a closed connected orientable $1$-manifold.
This is a diagram of the Klein bottle, note that the diagonal lines divide it into 2 Möbius strips sharing a boundary:
So the answer is yes.
Best Answer
From the comments above, we can do as follows. Note that the Klein bottle can be obtained by gluing two copies of the Mobius band along their boundaries. Hence by the formula in the question we have $$\chi(Klein)=2\chi(Mobius)-\chi(S^1).$$ Now observe that the Mobius band has the same homotopy type as the circle $S^1$, so they have the same Euler characteristic, which is 0. Therefore, the Euler characteristic of the Klein bottle is also 0.