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date: 2022-04-01 09:19
modification date: Friday 1st April 2022 09:19:53
title: "Lefschetz hyperplane theorem"
aliases: [Lefschetz hyperplane theorem]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [hard Lefschetz](Unsorted/weak%20and%20hard%20Lefschetz%20theorems.md)
	- [Morse theory](Morse%20theory.md)

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# Lefschetz hyperplane theorem


Idea: for $Y$ a [hyperplane section](hyperplane%20section) in $X$ with smooth complement, the homotopy type of $X$ determines that of $Y$. 

The statement: $(X, Y)$ is relatively $(n-1)\dash$connected, i.e. the [relative homotopy groups](relative%20homotopy%20groups) $\pi_k(X, Y)$ are zero in degrees $k\leq n-1$. Equivalently, the inclusion $Y\injects X$ is $(n-2)\dash$connected and a surjection on $\pi_{n-1}$.

Equivalent statements replace $\pi_k$ with $H^n$ or $H_n$.

![](attachments/Pasted%20image%2020220401094633.png)