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examples of cohomology rings
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\begin{align*}H^*(S^n) = {\mathbf{Z}}[x] / (x^2) = { {\bigwedge}^{\scriptscriptstyle \bullet}} _{\mathbf{Z}} { \left[ \scriptstyle {x} \right] } , \qquad {\left\lvert {x} \right\rvert} = n.\end{align*}
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\begin{align*}H^{*}\left(B C_{n}\right)=\mathbb{Z}[x] /(n x), \quad|x|=2 .\end{align*}
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\begin{align*}H^*({\mathbf{CP}}^n) = {\mathbf{Z}}[x]/(x^n), \qquad {\left\lvert {x} \right\rvert} = 2\end{align*}
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cohomology of infinite complex projective space \begin{align*}H^*({\mathbf{CP}}^\infty; {\mathbf{Z}}) = {\mathbf{Z}}[x], \qquad {\left\lvert {x} \right\rvert} = 2\end{align*}
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\begin{align*}H^*({\mathbf{RP}}^n) = { \mathbf{F} }_2[x]/(x^{n+1}), \qquad {\left\lvert {x} \right\rvert} = 1\end{align*}
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\begin{align*}H^*((S^1){ {}^{ \scriptscriptstyle\times^{n} } }) = { {\bigwedge}^{\scriptscriptstyle \bullet}} _{\mathbf{Z}} { \left[ \scriptstyle {x_1, \cdots, x_n} \right] } , \qquad {\left\lvert {x_k} \right\rvert} = 1\end{align*}
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\(K({\mathbf{Z}}, 1)\):
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\(K({\mathbf{Z}}, 3)\):
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\begin{align*}H^*(K({\mathbf{Z}}, 2n); {\mathbf{Q}}) = { {\bigwedge}^{\scriptscriptstyle \bullet}} _{\mathbf{Q}}[i_{2n}], \qquad {\left\lvert {i_{2n}} \right\rvert} = ?\end{align*}
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- \begin{align*}H^*(K({\mathbf{Z}}, 2n+1); {\mathbf{Q}}) = {\mathbf{Q}}[i_{2n+1}], \qquad {\left\lvert {i_{2n+1}} \right\rvert} = ?\end{align*}
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\begin{align*}H^{*}\left(B O(n) ; \mathbb{F}_{2}\right)=\mathbb{F}_{2}\left[w_{1}, \ldots, w_{n}\right], \quad w_{1} \in H^{1}\left(B ; \mathbb{F}_{2}\right)\end{align*}
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Cohomology of BSO. Spoiler: \begin{align*}H^*({{\mathbf{B}}{\operatorname{SO}}}_{2n+1}; C_2) \cong C_2 { \left[ \scriptstyle {w_2, w_3, \cdots, w_{2n+1}} \right] } , \qquad {\left\lvert {w_i} \right\rvert} = ?\end{align*}
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Cohomology of loop spaces of spheres. Spoiler: \begin{align*}{ {H}_{\scriptscriptstyle \bullet}} ({\Omega}S^n; {\mathbf{Z}}) = \bigoplus_{q\equiv 0 \operatorname{mod}(n-1)}{ \Sigma^{\scriptstyle[q]} {\mathbf{Z}} }\end{align*}
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cohomology of unitary groups. Spoler:
\begin{align*}H_*({\operatorname{U}}_n({\mathbf{R}}); {\mathbf{Z}}) = { {\bigwedge}^{\scriptscriptstyle \bullet}} { \left[ \scriptstyle {s_1,s_3,\cdots, s_{2n-1}} \right] } \qquad {\left\lvert {s_{2k-1}} \right\rvert} = 2k-1\end{align*}
- Note that \({\operatorname{U}}_n({\mathbf{C}}) = \operatorname{GL}_n({\mathbf{C}})\).
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cohomology of unitary groups. Spoler:
\begin{align*}H_*({\operatorname{U}}_n({\mathbf{R}}); {\mathbf{Z}}) = { {\bigwedge}^{\scriptscriptstyle \bullet}} { \left[ \scriptstyle {s_1,s_3,\cdots, s_{2n-1}} \right] } \qquad {\left\lvert {s_{2k-1}} \right\rvert} = 2k-1\end{align*}
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cohomology of lens spaces \begin{align*} H^*(L(p, q); {\mathbf{Z}}) = { \Sigma^{\scriptstyle[0]} {\mathbf{Z}} } \bigoplus_{1\leq k \leq n}{ \Sigma^{\scriptstyle[2k]} C_q }\oplus { \Sigma^{\scriptstyle[2p+1]} {\mathbf{Z}} }\end{align*}
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\begin{align*}H^*(L(\infty, q); {\mathbf{Z}}) = H^*(K(C_q, 1);{\mathbf{Z}}) = { \Sigma^{\scriptstyle[0]} {\mathbf{Z}} } \oplus \bigoplus_{k\geq 1}{ \Sigma^{\scriptstyle[2k]} C_q }\end{align*}
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\begin{align*}H^*({\mathbf{RP}}^\infty; {\mathbf{Z}}) = H^*(K(C_2, 1); {\mathbf{Z}}) = { \Sigma^{\scriptstyle[0]} {\mathbf{Z}} } \oplus \bigoplus_{k\geq 1}{ \Sigma^{\scriptstyle[2k]} C_2 }\end{align*}
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Unitary groups: use \({\operatorname{U}}_{n-1} \to {\operatorname{U}}_n \to S^{2n-1}\). \begin{align*}H^*({\operatorname{U}}_n; {\mathbf{Z}}) = { {\bigwedge}^{\scriptscriptstyle \bullet}} _{\mathbf{Z}} { \left[ \scriptstyle {x_1,x_3,\cdots, x_{2n-1}} \right] } \qquad {\left\lvert {x_k} \right\rvert} = k\end{align*}
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Special unitary groups uses \({\operatorname{SU}}_{n-1} \to {\operatorname{SU}}_n \to S^{2n-1}\) and \({\operatorname{SU}}_2\cong S^3\); \begin{align*} H^*({\operatorname{SU}}_n; {\mathbf{Z}}) = { {\bigwedge}^{\scriptscriptstyle \bullet}} _{\mathbf{Z}} { \left[ \scriptstyle {x_3,x_5,\cdots, x_{2n-1}} \right] } \qquad {\left\lvert {x_k} \right\rvert} = k \end{align*}
Exercises
