Compute the simplicial homology groups of the Klein bottle using the \(\Delta\)-complex structure described at the beginning of this section.
Solution
Use the standard square model of the Klein bottle with all four vertices identified, and draw the diagonal from the lower-left vertex to the upper-right vertex. Let \(a\) be the horizontal edge class, \(b\) the vertical edge class, and \(c\) the diagonal. Let \(U,L\) denote the two oriented \(2\)-simplices, with orientations chosen so that \[ \partial U=a+b-c, \qquad \partial L=a-b-c. \]
1. The simplicial chain complex is\[ 0\longrightarrow \mathbb Z^2 \xrightarrow{\partial_2} \mathbb Z^3 \xrightarrow{0} \mathbb Z \longrightarrow0, \]where, in the ordered bases \((U,L)\) and \((a,b,c)\),\[ [\partial_2] = \begin{pmatrix} 1&1\\ 1&-1\\ -1&-1 \end{pmatrix}. \]
There are two triangles, three edge classes, and one vertex. Thus \(C_2=\mathbb Z^2\), \(C_1=\mathbb Z^3\), and \(C_0=\mathbb Z\). All edge endpoints are the unique vertex, so \(\partial_1=0\). Tracing the three oriented sides of the two triangles in the square model gives the displayed formulas for \(\partial U\) and \(\partial L\).
2. The map \(\partial_2\) is injective.
If \[ x(a+b-c)+y(a-b-c)=0, \] then the \(b\)-coordinate gives \(x-y=0\), while the \(a\)-coordinate gives \(x+y=0\). Hence \(x=y=0\).
3. The quotient\[ C_1/\operatorname{im}\partial_2 \]is isomorphic to\[ \mathbb Z\oplus\mathbb Z/2. \]
The two boundary relations are \[ a+b-c=0, \qquad a-b-c=0. \] Subtracting them gives \[ 2b=0. \] The first relation then expresses \[ a=c-b. \] Thus the quotient is generated freely by \(c\) together with \(b\) of order two: \[ C_1/\operatorname{im}\partial_2 \cong \langle c\rangle\oplus\langle b\mid2b=0\rangle \cong \mathbb Z\oplus\mathbb Z/2. \]
4. Hence the simplicial homology groups of the Klein bottle are\[ \boxed{ H_0\cong\mathbb Z, \qquad H_1\cong\mathbb Z\oplus\mathbb Z/2, \qquad H_k=0\quad(k\ge2). } \]
3.Injectivity of \(\partial_2\) gives \(H_2=0\), and there are no higher simplices.