Compute the homology groups with local coefficients \(H_n(M; M_{\mathbb{Z}})\) for a closed nonorientable surface \(M\).
Solution
1. Setup and Poincaré Duality with local coefficients:
1.1. Let \(M = N_g\) (\(g \ge 1\)) be a closed connected nonorientable surface of genus \(g\) (homeomorphic to \(\#^g \mathbb{RP}^2\)).Let \(\mathcal{L} = \mathcal{M}_\mathbb{Z}\) be the orientation local coefficient system with fiber \(\mathbb{Z}\), where loops act by \(\pm 1\) according to their orientation character \(w_1: \pi_1(M) \to \{\pm 1\}\).
1.2. By Poincaré Duality for manifolds with local coefficients, for any closed \(n\)-manifold:\[ H_k(M; \mathcal{L}) \cong H^{n-k}(M; \mathbb{Z}). \]For \(n = 2\), this gives \(H_k(M; \mathcal{L}) \cong H^{2-k}(M; \mathbb{Z})\).
2. Cohomology of \(M\) with standard \(\mathbb{Z}\) coefficients:
2.1. The integral homology groups of \(M = N_g\) are:- \(H_0(M; \mathbb{Z}) \cong \mathbb{Z}\),- \(H_1(M; \mathbb{Z}) \cong \mathbb{Z}^{g-1} \oplus \mathbb{Z}_2\),- \(H_2(M; \mathbb{Z}) \cong 0\),- \(H_k(M; \mathbb{Z}) = 0\) for \(k > 2\).
2.2. Applying the Universal Coefficient Theorem for Cohomology \(H^k(M) \cong \operatorname{Hom}(H_k(M), \mathbb{Z}) \oplus \operatorname{Ext}(H_{k-1}(M), \mathbb{Z})\):- \(H^0(M; \mathbb{Z}) \cong \operatorname{Hom}(\mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z}\),- \(H^1(M; \mathbb{Z}) \cong \operatorname{Hom}(\mathbb{Z}^{g-1} \oplus \mathbb{Z}_2, \mathbb{Z}) \oplus \operatorname{Ext}(\mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z}^{g-1}\),- \(H^2(M; \mathbb{Z}) \cong \operatorname{Hom}(0, \mathbb{Z}) \oplus \operatorname{Ext}(\mathbb{Z}^{g-1} \oplus \mathbb{Z}_2, \mathbb{Z}) \cong 0 \oplus \mathbb{Z}_2 \cong \mathbb{Z}_2\),- \(H^k(M; \mathbb{Z}) = 0\) for \(k > 2\).
3. Computation of \(H_n(M; \mathcal{L})\):
3.1. Applying the Poincaré Duality isomorphisms \(H_k(M; \mathcal{L}) \cong H^{2-k}(M; \mathbb{Z})\):\[ H_n(M; \mathcal{M}_\mathbb{Z}) \cong \begin{cases} \mathbb{Z} & n = 2, \\ \mathbb{Z}^{g-1} & n = 1, \\ \mathbb{Z}_2 & n = 0, \\ 0 & n \ge 3 \text{ or } n < 0. \end{cases} \]
4. Verification via the twisted cellular chain complex:
4.1. In the standard CW structure of \(N_g\) (one 0-cell, \(g\) 1-cells \(a_1, \dots, a_g\), and one 2-cell attached along \(a_1^2 \cdots a_g^2\)), the orientation twist gives:- \(d_1(a_i) = (-1 - 1) e^0 = -2 e^0\), so \(\operatorname{Im}(d_1) = 2\mathbb{Z}\), yielding \(H_0 \cong \mathbb{Z} / 2\mathbb{Z} \cong \mathbb{Z}_2\).- \(d_2(e^2) = \sum_{i=1}^g (1 + (-1)) a_i = 0\), so \(\ker(d_2) = \mathbb{Z}\), yielding \(H_2 \cong \mathbb{Z}\).- \(H_1 \cong \ker(d_1) / \operatorname{Im}(d_2) = \mathbb{Z}^{g-1} / 0 \cong \mathbb{Z}^{g-1}\).
5. Conclusion:\(H_2(M; \mathcal{M}_\mathbb{Z}) \cong \mathbb{Z}\), \(H_1(M; \mathcal{M}_\mathbb{Z}) \cong \mathbb{Z}^{g-1}\), \(H_0(M; \mathcal{M}_\mathbb{Z}) \cong \mathbb{Z}_2\), and \(H_n(M; \mathcal{M}_\mathbb{Z}) = 0\) for \(n \ge 3\). Q.E.D.