Compute the integer homology groups \(H_*(\mathbb{RP}^2; \mathbb{Z})\) using the Mayer–Vietoris sequence by decomposing \(\mathbb{RP}^2\) as the union of a Möbius strip \(M\) and a 2-disk \(D^2\) along their boundary circle \(\partial M = S^1\).
Solution
Write \[ \mathbb{RP}^2=M\cup_{S^1}D^2, \] where \(M\) is a Möbius band and the attaching map identifies \(\partial D^2\) with \(\partial M\). Use Mayer–Vietoris for this excisive CW decomposition (equivalently, replace \(M\) and \(D^2\) by open neighborhoods deformation retracting onto them whose intersection deformation retracts onto the common boundary circle).
1. The relevant homology groups are\[ H_1(S^1)\cong\mathbb Z, \qquad H_1(M)\cong\mathbb Z, \qquad H_1(D^2)=0, \]and \(H_2\) of all three pieces is zero.
2. Under a deformation retraction of \(M\) onto its core circle, the boundary circle of the Möbius band winds twice around the core. Hence the inclusion\[ i:S^1=\partial M\hookrightarrow M \]induces multiplication by \(2\) on \(H_1\). The inclusion into \(D^2\) induces the zero map on \(H_1\).
3. The Mayer–Vietoris sequence in degrees \(2\) and \(1\) is therefore\[ 0\longrightarrow H_2(\mathbb{RP}^2) \longrightarrow \mathbb Z \xrightarrow{\,2\,}\mathbb Z \longrightarrow H_1(\mathbb{RP}^2) \longrightarrow \mathbb Z \xrightarrow{(1,-1)}\mathbb Z\oplus\mathbb Z. \]
3.1. Multiplication by \(2\) is injective, so exactness gives\[ H_2(\mathbb{RP}^2)=0. \]
3.2. The map \((1,-1)\) is injective, so the preceding connecting map is zero. Hence\[ H_1(\mathbb{RP}^2) \cong\operatorname{coker}(\mathbb Z\xrightarrow{2}\mathbb Z) \cong\mathbb Z/2. \]
4. Since \(\mathbb{RP}^2\) is connected,\[ H_0(\mathbb{RP}^2)\cong\mathbb Z, \]and all homology groups above degree \(2\) vanish. Thus\[ H_k(\mathbb{RP}^2;\mathbb Z)\cong \begin{cases} \mathbb Z,&k=0,\\ \mathbb Z/2,&k=1,\\ 0,&k\ge2. \end{cases} \]