----- > [!proposition] Proposition. ([[the Euler characteristic of an odd-dimensional compact manifold is zero]]) > Let $M$ be an odd-dimensional [[compact]] [[manifold]].[^1] Then the [[the Euler characteristic of an odd-dimensional compact manifold is zero|Euler characteristic]] of $M$ is zero: $\chi(M)=0$. ^proposition > [!proof]- Proof. ([[the Euler characteristic of an odd-dimensional compact manifold is zero]]) > Say $\text{dim }M=2n+1$. > > Pick $R=\mathbb{F}_2$. Then $M$ ([[(homological) orientation of a manifold|like all]] [[manifold|manifolds]]) is $\mathbb{F}_{2}$-oriented. Since we can compute Euler characteristic using coefficients [[field|in]] $\mathbb{F}_2$, have $\chi(M)=\sum_{r=0}^{2n+1}(-1)^{r} \text{dim}_{\mathbb{F}_{2}}H_{i}(M; \mathbb{F}_{2}).$ > But we know $H_{i}(M; \mathbb{F}_{2})\cong_{\text{PD}} H^{2n+1-i}(M; \mathbb{F}_{2}) \cong_{\text{UCT}} \text{Hom}_{\mathbb{F}_{2}\text{-}\mathsf{Mod}}\big(H_{2n+1-i}(M; \mathbb{F}_{2}) , \mathbb{F}_{2}\big)$ > by [[Poincare duality]] and the [[universal coefficients theorem for cohomology|universal coefficients theorem]]. This latter space is in turn isomorphic (via [[dual vector space|dual space]] considerations) to to $H_{2n+1-i}(M; \mathbb{F}_{2})$. So $\text{dim}_{\mathbb{F}_{2}}H_{i}(M; \mathbb{F}_{2})=\text{dim}_{\mathbb{F}_2}H_{2n+1-i}(M; \mathbb{F}_{2}).$ > But these show up in the sum above with opposite signs, so they cancel. The sum is therefore zero. ----- #### [^1]: Technically, with our definitions and results thus far we have to add the assumption '$M$ is [[homotopy equivalent]] to a finite [[cell complex]]'. This is 'not a real requirement'. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```