---- > [!definition] Definition. ([[Gysin sequence]]) > Let $X$ be a [[topological space]] and $E \xrightarrow{\pi}X$ be a $d$-dimensional [[vector bundle]]. Put $E^{\sharp}:=E-s_{0}(X)$. > > Suppose further that $E$ is $R$-[[orientation of a vector bundle|oriented]] for some (say, [[commutative ring|commutative]]) [[ring]] $R$. From the [[long exact sequence for relative singular homology|LES]] for the [[topological pair|pair]] $(E,E^{\sharp})$ on [[singular cohomology|cohomology]] protrudes [[isomorphism|isomorphisms]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BjKCHBAL6l0mXPkIoAjOSq1GLNgAkAesCwBqKAIAUAUVIACXcs5wAFnQBOaANwGASgEoQQkdjwEiAJhnV6zVkQQFTVNHV07JxdhEAx3cSIAZl85AKVVDS09E3ZzK1so11jRDwlkABYU-wUgkMz1SXDSY1MLaxtCmLixTxQAVir5QI5uXn5otx6y6UlZauG6nQANDuci7tLvUlm-IfTQrJWHNa6ShJRkndSa4Iyw7U4AWzocMwAjN+AAZXDHVYnivFehVtnM9rU7o1tCtCrIYFAAObwIigABmlggjyQZBAOAgSEaMXRmKQlVx+MQfSKxKxiGk5KQPmuwwAjsoAFQAmmM6h4pDJZlsTj4HB0DlcjG0gV8xBk+ZC9hoKx4Rji6mSpAAdl5FKZ8qCnDQWDVRI1iCZMu1gqCcAA+sQTWizQA2HUE3ZpA3sAAKZiwEpJiAAnG7ZR6bpxff7qAw6G8YAxvWdeiBLFgEWYcAGpaGABzh4YAK3FMbjCaTQIkqfTmZcFAEQA > \begin{tikzcd} > \cdots \arrow[r] & {H^{i+d}(E, E^\sharp; R)} \arrow[r, "q^*"] & H^{i+d}(E; R) \arrow[r, "\iota^*"] \arrow[d, "s_0^*"] & H^{i+d}(E^\sharp;R) \arrow[r, "\partial^*"] \arrow[d, "j^*"'] & {H^{i+d+1}(E,E^\sharp;R)} \arrow[r] & \cdots \\ > & H^{i}(X;R) \arrow[u, "\Phi"] & H^{i+d}(X; R) \arrow[u, "\pi^*"] & H^{i+d}(\mathbb{S}(E);R) & H^{i+1}(X;R) \arrow[u, "\Phi"'] & > \end{tikzcd} > \end{document} > ``` > > where: > 1. $\Phi$ is [[The Thom isomorphism theorem|Thom isomorphism]]; > 2. $\pi^{*}$ and $s_{0}^{*}$ are the ([[homotopy invariance of singular homology|mutual inverse]]) [[singular (co)chain map and homomorphism induced by a continuous map|maps induced on]] [[(co)homology of a complex|cohomology]] by the [[homotopy equivalent|homotopy inverses]] $\pi,s_{0}$; > 3. $j^{*}$ is the [[homomorphism on cohomology induced by a cochain map|map induced]] by the [[homotopy equivalent|homotopy equivalence]] $\mathbb{S}(E) \hookrightarrow E^{\sharp}$. > > Following the vertical arrows 'up and around' yields a [[exact sequence|long exact sequence]] $\cdots \to H^{i}(X;R) \xrightarrow{s_{0}^{*}q^{*}\Phi}H^{i+d}(X;R) \xrightarrow{j^{*}\iota^{*}\pi^{*}} H^{i+d}\big( \mathbb{S}(E);R \big) \xrightarrow{\Phi ^{-1}\partial^{*}j^{*^{-1}}}H^{i}(X;R) \to \cdots$ > The first two of these maps can be explicitly described as $- \smile e(E)$ and $p^{*}$ respectively, where $p:\mathbb{S}(E) \to X$ is the projection $\pi |_{\mathbb{S}(E)}$. The third is denoted $p_{!}$. Putting everything together yields the [[diagram]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BjKCHBAL6l0mXPkIoAjOSq1GLNgAkAesCwBqKAIAUAUVIACXcs5wAFnQBOaANwGASgEoQQkdjwEiAJhnV6zVkQQFTVNHV07JxdhEAx3cSIAZl85AKVVDS09E3ZzK1so11jRDwlkABYU-wUgkMz1SXDSY1MLaxtCmLixTxQAVir5QI5uXn5otx6y6UlZauG6nQANDuci7tLvUlm-IfTQrJWHNa6ShJRkndSa4Iyw7U4AWzocMwAjN+AAZXDHVYnivFehVtnM9rU7o1tCtOpNNv1Qbs0kFODw+IJTkCymQrvM2KixoJZDAoABzeBEUAAM0sEEeSDIIBwECQjRiNLpSEqTJZiD6RQ59MQ0h5SB812GAEdlAAqAGCsXUZlIZIS-HsfA4Oiy+W0oWq5WIbl4lHsNBWPCMHUCvVIADsSt54pNIzQWGt7NtiHFhodaqCcAA+sQPdSvQA2R2spE3TgABTMWF1nMQAE4o0aY8N44mQNQGHQ3jAGHGzr0QJYsKSzDhk-qMwAOLNsABWOvzheLpaxbEr1drNpTkdFiD9LoAtAZTI8sAwYAYYHpnB2iyWyxIK1Wa3X7Y3m0E0O2QAXV92pr2twPPSmmyP0-7YoGAIR54+dtc9oJ97eDoX3w2SMQv6siKhrhgIFACEAA > \begin{tikzcd} > \cdots \arrow[r] & {H^{i+d}(E, E^\sharp; R)} \arrow[r, "q^*"] & H^{i+d}(E; R) \arrow[r, "\iota^*"] \arrow[d, "s_0^*"] & H^{i+d}(E^\sharp;R) \arrow[r, "\partial^*"] \arrow[d, "j^*"'] & {H^{i+d+1}(E,E^\sharp;R)} \arrow[r] & \cdots \\ > \cdots \arrow[r] & H^{i}(X;R) \arrow[u, "\Phi"] \arrow[r, "- \smile e(E)"'] & H^{i+d}(X; R) \arrow[u, "\pi^*"] \arrow[r, "p^*"'] & H^{i+d}(\mathbb{S}(E);R) \arrow[r, "p_!"'] & H^{i+1}(X;R) \arrow[u, "\Phi"'] \arrow[r] & \cdots > \end{tikzcd} > \end{document} > ``` > > > The [[exact sequence]] in the bottom row is called the **Gysin sequence** of the [[sphere bundle]] $\mathbb{S}(E) \to X$. It depends on the choice of $R$-[[orientation of a vector bundle|orientation]] of $E$. > The [[relative cup product]] makes the Gysin sequence into an [[exact sequence]] of $H^{*}(X;R)$-[[module|modules]] (because the top LES is as such). > [!justification] > Need to verify that the two maps [[cup product|equal]] $- \smile e(E)$ and $p^{*}$ as claimed. > For a given class $\alpha \in H^{i}(X;R)$: $\begin{align} (s_{0}^{*}q^{*}\Phi)(\alpha) &= s_{0}^{*}q^{*}\big( \pi^{*}(\alpha) \smile u_{E} \big) \\ &= s_{0}^{*} \big( q^{*}\pi^{*}(\alpha) \smile q^{*}u_{E} \big) \\ &= s_{0}^{*}\pi^{*}(\alpha) \smile s_{0}^{*}q^{*}u_{E} \\ &= \alpha \smile e(E), \end{align}$ where [[covariant functor|functoriality]] of the [[cup product]] has been employed twice, we have used $q^{*}\pi^{*}\alpha=\pi^{*}\alpha$ (intuitively, $q^{*}$ 'forgets we're in [[relative singular homology|relative cohomology]], but $\pi^{*}\alpha$ is already not a relative class), and we have used $s_{0}^{*}=\pi^{* ^{-1}}$. > As for why $p^{*}=j^{*}\iota^{*} \pi^{*}$, this is clear because [[singular (co)chain map and homomorphism induced by a continuous map|singular cohomology is a]] [[contravariant functor|(contravariant) functor]]: $j^{*}\iota^{*}\pi^{*}=(\pi \circ \iota \circ j)^{*}$, and $\pi \circ \iota \circ j=\mathbb{S}(E) \xhookrightarrow{j} E^{\sharp} \xhookrightarrow{\iota} E \xrightarrow{\pi}X$ is precisely the restriction of $\pi$ to $\mathbb{S}(E)$, i.e., $p$. ^justification > [!basicexample] Example. (The cohomology ring for $\mathbb{C}P^{n}$) > > Let $L=(\gamma_{1,n+1}^{\mathbb{C}} \to \mathbb{C}P^{n})$ be the [[tautological vector bundle|tautological complex line bundle]], $L=\{ (v, V): v \in \mathbb{C}P^{n} , v \in V \} \to \mathbb{C}P^{n}$. Inherits the Hermitian [[inner product]] from $\mathbb{C}^{n+1}$; the consequent [[sphere bundle]] $\mathbb{S}(L)=\{ (v, V): \langle v,v \rangle=1 \}$ is [[homeomorphism]] to $\mathbb{S}^{2n+1}$. $L$ has a canonical $\mathbb{Z}$-orientation (like every complex vector bundle does). So Gysin sequence may be applied (starting on the right): $\dots \to H^{i+1}(\mathbb{S}^{2n+1}) \xrightarrow{p_{!}} H^{i}(\mathbb{C}P^{n}) \xrightarrow{- \smile e(E)} H^{i+2}(\mathbb{C}P^{n}) \xrightarrow{p^{*}} H^{i+2}(\mathbb{S}^{2n+1}) \to \cdots $ > This can't be that hard, there are a bunch of spheres... > > $\dots \to \underbrace{ H^{i+1}(\mathbb{S}^{2n+1}) }_{ =0, 0 \leq i < 2n } \xrightarrow{p_{!}} H^{i}(\mathbb{C}P^{n}) \xrightarrow{- \smile e(E)} H^{i+2}(\mathbb{C}P^{n}) \xrightarrow{p^{*}} \underbrace{ H^{i+2}(\mathbb{S}^{2n+1}) }_{ =0, 0 \leq i < 2n-1 } \to \cdots .$ > So $- \smile e(L):H^{i}(\mathbb{C}P^{n}) \to H^{i+2}(\mathbb{C}P^{n})$ is an [[isomorphism]] for $i \leq 2n-2$. > > Checking what happens at the endpoints carefully, the conclusion is that $H^{*}(\mathbb{C}P^{n})=\mathbb{Z}[e(L)] / (e(L)^{n+1})$ as a ring, with $e(L)=1 \smile e(L)$ a generator in degree $2$ (see handwritten notes too). > [!basicexample] Example. (The cohomology ring for $\mathbb{R}P^{n}$ with $\mathbb{F}_{2}$ coefficients) > Consider $E=$ $\gamma_{1,n+1}^{\mathbb{R}} \to \mathbb{R}P^{n}$. Has rank $d=1$. Orient with $\mathbb{F}_{2}$ coefficients. $E$ inherits from $\mathbb{R}^{n+1}$ an inner product, with respect to which the sphere bundle $\mathbb{S}(E)$ is given by $\mathbb{S}^{n}$. Now running the Gysin sequence from one $p_{!}$ to the next $p_{!}$ : $\dots \to \underbrace{ H^{i+1}(\mathbb{S}^{n}; \mathbb{F}_{2}) }_{ =0, 0 \leq i \leq n-1 } \xrightarrow{p_{!}} H^{i}(\mathbb{R}P^{n}; \mathbb{F}_{2}) \xrightarrow{ - \smile e(E)} H^{i+1}(\mathbb{R}P^{n};\mathbb{F}_{2}) \xrightarrow{p^{*}} \underbrace{ H^{i+2}(\mathbb{S}^{n}; \mathbb{F}_{2}) }_{ =0, 0 \leq i \leq n-2 }\to \dots$ > we see that for $0 \leq i \leq n-2$, $- \smile e(E):H^{i} \to H^{i+1}$ is an isomorphism. So with $1$ the generator in degree $0$ of $H^{*}(\mathbb{C}P^{n})$, we have $e(E)=1 \smile e(E)$ generating degree 1, $e(E) ^{2}$ generating degree $2$, and so on for $e(E)^{i}$ generating $H^{i}$ up until $i=n$, when $\smile e(E)=0$. > > The conclusion is that $H^{*}(\mathbb{R}P^{n}; \mathbb{F}_{2})=\frac{\mathbb{F}_{2}[e(E)]}{e(E)^{n+1}}.$ > > > ---- #### ---- #### 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 > ```