---- > [!definition] Definition. ([[diagonal class of a manifold]]) > > Let $M$ be a [[smooth manifold|smooth]] $(\mathbb{Z}\text{-})$ [[(homological) orientation of a manifold|oriented]] [[compact]] [[manifold]] of dimension $d$. Let $f:M \to M$ be a [[smooth maps between manifolds|smooth map]]. Denote by $\Delta(M)$ the [[Hausdorff iff diagonal is closed|diagonal]] of $M$, $\Delta(M):=\{ (x, x): x \in M \} \subset M \times M$; > > Viewing $\Delta(M)$ as a $d$-dimensional [[embedded submanifold|submanifold]] with [[The Thom Theorem for oriented manifolds|fundamental class]] $[\Delta(M)] \in H_{d}(M \times M; \mathbb{Q})$, denote by $\delta=D_{M} ^{-1}([\Delta(M)]) \in H^{d}(M \times M; \mathbb{Q})$ its [[Poincare duality|Poincare dual]]. We can call $\delta$ the **diagonal class** of $M$. > > Explicitly, $\delta$ may be computed as follows. Let $\{ a_{i} \}_{i \in I}$ be [[basis]] of the [[graded module|graded]] $\mathbb{Q}$-[[vector space]] $H^{*}(M;\mathbb{Q})$ and $\{ b_{i} \}_{i \in I}$ its [[dual basis]] wrt the [[the perfect Poincare pairing|Poincare pairing]][^1] $\langle -,- \rangle:H^{k}(M;\mathbb{Q}) \otimes H^{d-k}(M; \mathbb{Q}) \to \mathbb{Q}$. The set $\{ a_{i} \otimes b_{i} \}$ [[when is an element of a tensor product nonzero?|is then a]] [[basis]] for $H^{*}(M; \mathbb{Q}) \otimes H^{*}(M; \mathbb{Q})$, [[Kunneth Theorem for singular cohomology|which recall]] is the [[graded module|graded vector space]] which in grade $n$ is given by $\bigoplus_{k+ \ell=n}H^{k}(M; \mathbb{Q}) \otimes H^{\ell}(M; \mathbb{Q})\underbrace{ = }_{ \text{Kunneth} }H^{n}(M \times M; \mathbb{Q}).$ > Then $\begin{align} > \delta&=\sum_{i \in I} (-1)^{|b_{i}| } \ a_{i} \otimes b_{i} \in \underbrace{ H^{d}(M \times M; \mathbb{Q}) }_{ \bigoplus_{k+\ell=d}H^{k}(M; \mathbb{Q}) \otimes H^{\ell}(M; \mathbb{Q}) }. > \end{align}$ > Notably, the [[singular (co)chain map and homomorphism induced by a continuous map|pullback]] of $\delta$ under the diagonal embedding $\Delta:M \to M \times M$ evaluates against the [[The Thom Theorem for oriented manifolds|fundamental class]] $[M]$ to the [[Euler characteristic of a cell complex|Euler characteristic]]: $\Delta^{*}(\delta)([M])=\chi(M).$ See more in [[Euler characteristic of a cell complex|Euler characteristic]]. $\delta$ also features in the [[Lefschetz fixed point theorem]]. ---- #### [^1]: [[the perfect Poincare pairing|Recall]] this is given by [[cup product|cupping]] then evaluating against the [[The Thom Theorem for oriented manifolds|fundamental class]] of $M$, i.e., $\langle \alpha, \beta \rangle=(\alpha \smile \beta)[M]$. Recall that, using [[Poincare duality]], we showed it is [[nondegenerate bilinear form|nondegenerate]] (hence the name) when [[universal coefficients theorem for homology|UCT]] applies, which it does because $\mathbb{Q}$ is a [[field]]. So $\langle a_{i}, b_{j} \rangle=\delta_{ij}$, and note that implicitly $\text{deg }a_{i}=d-\text{deg }b_{i}$ (otherwise they don't pair). ---- #### 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 > ``` ~~Viewing $\Delta(M)$ as a [[singular homology|singular homology class]] [[(co)homology with coefficients|with rational coefficients]], $[\Delta(M)] \in H_{0}(M \times M; \mathbb{Q})$, denote by $\delta=D_{M} ^{-1}([\Delta(M)]) \in H^{d}(M \times M; \mathbb{Q})$ its [[Poincare duality|Poincare dual]]. We can call $\delta$ the **diagonal class** of $M$. ~~