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  • hard Lefschetz
  • Morse theory

Lefschetz hyperplane theorem

Idea: for \(Y\) a hyperplane section in \(X\) with smooth complement, the homotopy type of \(X\) determines that of \(Y\).

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

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