---- > [!definition] Definition. ([[Euler characteristic of a cell complex]]) > Let $X$ be a [[cell complex]]. The following integers are all equal: > 1. $\chi (X):= \sum_{n}(-1)^{n} \text{ number of }n\text{-cells of }X$ > 2. $\chi_{\mathbb{Z}}(X):=\sum_{n}(-1)^{n} \text{ rank }H_{n}(X; \mathbb{Z})$ > 3. $\chi_{\mathbb{F}}(X):= \sum_{n}(-1)^{n} \ \text{dim}_{\mathbb{F}}H_{n}(X; \mathbb{F})$ for $\mathbb{F}$ a [[field]] > > $\chi(X)$ is called the **Euler characteristic** of $X$. > [!proposition] For (nice) manifolds. > If $X=M$ is in fact a [[smooth manifold]] of dimension $d$ that is [[compact]], then $\chi(M)$ takes even more equivalent forms. > > - If $d=2k+1$ is odd, [[the Euler characteristic of an odd-dimensional compact manifold is zero|then]] $\chi(M)=0$; > - If $M$ is ($\mathbb{Z}$-)[[(homological) orientation of a manifold|orientable]], then $\chi(M)$ is the evaluation of the [[singular (co)chain map and homomorphism induced by a continuous map|pullback]] of the [[diagonal class of a manifold|diagonal class]] $\delta$ under the diagonal embedding $\Delta:M \to M \times M$ against the [[The Thom Theorem for oriented manifolds|fundamental class]] $[M]$: $ \chi(M)=\Delta^{*}(\delta)([M]).$ > - If $M$ is ($\mathbb{Z}$-)[[(homological) orientation of a manifold|orientable]], then $\chi(M)$ is the evaluation of the [[The Thom isomorphism theorem|Euler class]] of the [[tangent bundle]] against the [[The Thom Theorem for oriented manifolds|fundamental class]]: $\chi(M)=e(TM) [M].$This connects the two appearances of Euler in our course. The Euler class is, in a sense, a more refined notion than Euler characteristic — intuitively, the latter only sees 'do [[vector field]]s vanish?' while the latter sees 'to what extent do vector fields vanish?'. We won't make this precise. ^proposition > [!basicexample] > - [[the Euler characteristic of an odd-dimensional compact manifold is zero]] > - [ ] definitely others ^basic-example > [!justification] Justification. ($\chi(X)=\chi_{\mathbb{Z}}(X)=\chi_{\mathbb{F}}(X)$) > (Recall that [[cellular homology|cellular]] and [[singular homology|singular]] [[cellular and singular homology agree|homology agree]]. This is essentially a rehashing of algebraic arguments also made elsewhere.) > > First note that the number of $n$-cells of $X$ is the [[rank of a free module|rank]] of $C_{n}^{\text{cell}}(X)$, which we'll just write as $C_{n}$. More notation: let $\begin{align} > Z_{n}&=\operatorname{ker }(d_{n}: C_{n}\to C_{n-1}) \\ > B_{n}& = \operatorname{im }(d_{n+1}:C_{n+1} \to C_{n}). > \end{align}$ > We are now going to write down two [[short exact sequence|short exact sequences]]. By definition of [[(co)homology of a complex|homology]], we have $0 \to B_{n} \hookrightarrow Z_{n} \twoheadrightarrow H_{n}(X; \mathbb{Z}) \to 0.$ > And the definition of $Z_{n}$ and $B_{n}$ gives $0 \to Z_{n} \to C_{n} \to B_{n-1} \to 0.$ > > > Applying to these (e.g.) the [[first isomorphism theorem]], we see that $\operatorname{rk} Z_{n}=\operatorname{rk} B_{n}+\operatorname{rk}H_{n}(X ;\mathbb{Z})$. It follows that $\chi_{\mathbb{Z}}(X)=\sum (-1)^{n} \operatorname{rk} H_{n}(X; \mathbb{Z})=\sum (-1)^{n} (\operatorname{rk}Z_{n}- \operatorname{rk}B_{n}).$Again using FIT we see $\operatorname{\operatorname{rk}}C_{n+1}=\operatorname{rk}B_{n}+\operatorname{rk}_{}Z_{n+1}$ and further deduce $\begin{align} \chi_{\mathbb{Z}}(X)&=\sum(-1)^{n} (\operatorname{rk}Z_{n}-\operatorname{rk}C_{n+1}+\operatorname{rk}Z_{n+1}) \\ &= \sum (-1)^{n+1} ( \operatorname{rk} C_{n+1} ) \\ &= \chi(X). \end{align}$ By using [[extension of scalars|the fact]] $\operatorname{rk}C_{n}=\text{dim}_{\mathbb{F}}C_{n} \otimes \mathbb{F}$ we obtain the result for $\chi_{\mathbb{F}}$. ---- #### - [[(co)homology with coefficients|homology with coefficients]] - [ ] page 46. also, relate to [[Euler characteristic of a complex of vector spaces]], or [[computing the Euler characteristic of a vector space complex using homology]] (or maybe not now) ---- #### 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 > ```