Talk:Betti numbers of 3-manifolds (Ex)
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Now let be a closed, orientable -manifold with . View as a finite CW complex, with cells in dimensions-, as usual. Attach cells of dimension to to obtain , a space. Then is the rank of and is the rank of . By construction of , and the rank of is bounded above by the rank of . Moreover, is a -manifold, so Poincaré duality says the ranks of and agree.
Thus, , so .
,1,2,3$, as usual. Attach cells of dimension $\ge 3$ to $M$ to obtain $N$, a $K(G,1)$ space. Then $b_1(G)$ is the rank of $H_1(N;\mathbb{Z})$ and $b_2(G)$ is the rank of $H_2(N;\mathbb{Z})$. By construction of $N$, $H_1(N;\mathbb{Z})\cong H_1(M;\mathbb{Z})$ and the rank of $H_2(N;\mathbb{Z})$ is bounded above by the rank of $H_2(M;\mathbb{Z})$. Moreover, $M$ is a $-manifold, so Poincaré duality says the ranks of $H_1(M;\mathbb{Z})$ and $H_2(M;\mathbb{Z})$ agree. Thus, $b_1(G)=\text{rk} H_1(N;\mathbb{Z})=\text{rk}H_1(M;\mathbb{Z})=\text{rk}H_2(M;\mathbb{Z})\ge \text{rk} H_2(N;\mathbb{Z})=b_2(G)$, so $b_1(G)\ge b_2(G)$. M be a closed, orientable -manifold with . View as a finite CW complex, with cells in dimensions-, as usual. Attach cells of dimension to to obtain , a space. Then is the rank of and is the rank of . By construction of , and the rank of is bounded above by the rank of . Moreover, is a -manifold, so Poincaré duality says the ranks of and agree.Thus, , so .