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date: 2022-04-03 20:14
modification date: Sunday 3rd April 2022 20:14:10
title: "examples of cohomology rings"
aliases: [examples of cohomology rings]

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# examples of cohomology rings

- $$H^*(S^n) = \ZZ[x] / (x^2) = \Extalg_\ZZ\adjoin{x}, \qquad \abs{x} = n.$$
- $$H^{*}\left(B C_{n}\right)=\mathbb{Z}[x] /(n x), \quad|x|=2 .$$
- $$H^*(\CP^n) = \ZZ[x]/(x^n), \qquad \abs{x} = 2$$
- [cohomology of infinite complex projective space](cohomology%20of%20infinite%20complex%20projective%20space.md)
  $$H^*(\CP^\infty; \ZZ) = \ZZ[x], \qquad \abs{x} = 2$$
- $$H^*(\RP^n) = \FF_2[x]/(x^{n+1}), \qquad \abs{x} = 1$$
- $$H^*((S^1)\cartpower{n}) = \Extalg_\ZZ\adjoin{x_1, \cdots, x_n}, \qquad \abs{x_k} = 1$$
- $K(\ZZ, 1)$: ![](attachments/Pasted%20image%2020220422204209.png)
- $K(\ZZ, 3)$: ![](attachments/Pasted%20image%2020220422204448.png)
- $$H^*(K(\ZZ, 2n); \QQ) = \Extalg_\QQ [i_{2n}], \qquad \abs{i_{2n}} = ?$$
- - $$H^*(K(\ZZ, 2n+1); \QQ) = \QQ [i_{2n+1}], \qquad \abs{i_{2n+1}} = ?$$
- $$H^{*}\left(B O(n) ; \mathbb{F}_{2}\right)=\mathbb{F}_{2}\left[w_{1}, \ldots, w_{n}\right], \quad w_{1} \in H^{1}\left(B ; \mathbb{F}_{2}\right)$$
- [Cohomology of BSO](Cohomology%20of%20BSO.md). Spoiler: $$H^*(\BSO_{2n+1}; C_2) \cong C_2\adjoin{w_2, w_3, \cdots, w_{2n+1}}, \qquad \abs{w_i} = ?$$
- [Cohomology of loop spaces of spheres](Cohomology%20of%20loop%20spaces%20of%20spheres.md). Spoiler: 
  $$\complex{H}(\Loop S^n; \ZZ) = \bigoplus_{q\equiv 0 \mod (n-1)}\shift{\ZZ}{q}$$
  - [cohomology of unitary groups](cohomology%20of%20unitary%20groups.md). Spoler: $$H_*(\U_n(\RR); \ZZ) =\Extalg\adjoin{s_1,s_3,\cdots, s_{2n-1}} \qquad \abs{s_{2k-1}} = 2k-1$$
	  - Note that $\U_n(\CC) = \GL_n(\CC)$.

- [cohomology of lens spaces](cohomology%20of%20lens%20spaces.md)
  $$ H^*(L(p, q); \ZZ) = \shift{\ZZ}{0}
 \bigoplus_{1\leq k \leq n}\shift{C_q}{2k}\oplus \shift{\ZZ}{2p+1}$$
 - $$H^*(L(\infty, q); \ZZ) = H^*(K(C_q, 1);\ZZ) = \shift{\ZZ}{0} \oplus \bigoplus_{k\geq 1}\shift{C_q}{2k}$$
 - $$H^*(\RP^\infty; \ZZ) = H^*(K(C_2, 1); \ZZ) = \shift{\ZZ}{0} \oplus \bigoplus_{k\geq 1}\shift{C_2}{2k}$$
 - [Unitary groups](https://people.math.wisc.edu/~maxim/spseq.pdf#page=33): use $\U_{n-1} \to \U_n \to S^{2n-1}$.
   $$H^*(\U_n; \ZZ) = \Extalg_\ZZ \adjoin{x_1,x_3,\cdots, x_{2n-1}} \qquad \abs{x_k} = k$$

- [Special unitary groups](https://people.math.wisc.edu/~maxim/spseq.pdf#page=34) uses $\SU_{n-1} \to \SU_n \to S^{2n-1}$ and $\SU_2\cong S^3$;
$$
H^*(\SU_n; \ZZ) = \Extalg_\ZZ \adjoin{x_3,x_5,\cdots, x_{2n-1}} \qquad \abs{x_k} = k
$$


# Exercises

![](attachments/Pasted%20image%2020220422212506.png)
![](attachments/Pasted%20image%2020220422212514.png)
![](attachments/Pasted%20image%2020220422212520.png)