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date: 2023-03-08 01:29
aliases: ["primitive cohomology"]
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Last modified: `=this.file.mday`

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# primitive cohomology

$$(\wait) \cup h^k: H^{n-k}(X) \rightarrow H^{n+k}(X),\qquad h\in H^2(X; \ZZ)$$
and 
$$H^{n-k}_\prim(X) = \ker\qty{(-) \cup h^{k+1}: H^{n-k}(X) \rightarrow H^{n+k+2}(X)}$$

Decomposes $H^{n-k}(X) = H^{n-k}_{\prim}(X) \oplus H^{n-k}_{\prim}(X)^\perp$ where $H^{n-k}_{\prim}(X)^\perp = \im \qty{(\wait)\cupprod h: H^{n-k-2}(X) \to H^{n-k}(X)}$ are non-primitive classes which are not induced by lower-dimensional classes.


![](attachments/2023-03-12prim.png)