If \(M_g\) denotes the closed orientable surface of genus \(g\), show that degree 1 maps \(M_g \to M_h\) exist iff \(g \geq h\).
Solution
1. Forward direction (\(f: M_g \to M_h\) of degree \(1 \implies g \ge h\)):
1.1. Let \(f: M_g \to M_h\) be a continuous map of degree 1.Then the induced map on top cohomology sends the fundamental class to the fundamental class:\[ f^*(\mu_h) = \mu_g, \quad \text{where } \mu_g \in H^2(M_g; \mathbb{R}) \cong \mathbb{R} \text{ and } \mu_h \in H^2(M_h; \mathbb{R}) \cong \mathbb{R}. \]
definition of degree via cohomology.
1.2. We claim \(f^*: H^1(M_h; \mathbb{R}) \to H^1(M_g; \mathbb{R})\) is injective:Let \(\alpha \in H^1(M_h; \mathbb{R}) \setminus \{0\}\).By Poincaré Duality on \(M_h\), the cup product pairing \(H^1(M_h; \mathbb{R}) \times H^1(M_h; \mathbb{R}) \to H^2(M_h; \mathbb{R})\) is non-degenerate (skew-symmetric and non-singular).Thus there exists \(\beta \in H^1(M_h; \mathbb{R})\) such that \(\alpha \smile \beta = \mu_h\).
Poincaré duality pairing on closed orientable surfaces.
1.3. Applying \(f^*\) and the ring homomorphism property of cup product:\[ f^*(\alpha) \smile f^*(\beta) = f^*(\alpha \smile \beta) = f^*(\mu_h) = \mu_g \neq 0. \]Because the cup product is non-zero, \(f^*(\alpha) \neq 0\).Thus \(\ker(f^*) = \{0\}\), so \(f^*: H^1(M_h; \mathbb{R}) \hookrightarrow H^1(M_g; \mathbb{R})\) is an injective linear map.
injectivity from non-vanishing cup product.
1.4. Comparing dimensions:\[ \dim_\mathbb{R} H^1(M_h; \mathbb{R}) \le \dim_\mathbb{R} H^1(M_g; \mathbb{R}) \implies 2h \le 2g \implies h \le g. \]
\(\dim H^1(M_k; \mathbb{R}) = 2k\).
2. Reverse direction (\(g \ge h \implies\) existence of degree 1 map \(M_g \to M_h\)):
2.1. If \(g = h\), the identity map \(\operatorname{id}: M_g \to M_g\) has degree 1.
\(\deg(\operatorname{id}) = 1\).
2.2. If \(g > h\), decompose \(M_g\) as the connected sum \(M_g = M_h \# M_{g-h}\).Let \(c\) be the separating circle along which the sum is glued.
connected sum decomposition of surfaces.
2.3. Define \(f: M_g \to M_h\) by collapsing the entire \(M_{g-h}\) summand (the closure of one component of \(M_g \setminus c\)) to a single point \(p \in M_h\), and mapping \(M_h \setminus \operatorname{int}(D^2)\) homeomorphically onto \(M_h \setminus \{p\}\).
pinch map collapsing one connected summand to a point.
2.4. Any point \(y \in M_h \setminus \{p\}\) is a regular value with \(f^{-1}(y) = \{x\}\) consisting of a single point where \(f\) is a local orientation-preserving homeomorphism.Thus \(\deg(f) = +1\).
local degree formula for degree of smooth/continuous maps.
3. Conclusion:Degree 1 maps \(M_g \to M_h\) exist if and only if \(g \ge h\). Q.E.D.
1 and 2.