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This is sade to be the Morse theory in Hilbert space ,and i want to know the definition (or where i can find it ) of :

  • The qth singular relative homology groupe

  • The qth critical group

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Please;

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A definition of relative homology is given for instance on page 115 of Hatcher's book on algebraic topology, which is freely available on his website.

The $q$-th critical group seems to be defined in your text itself as $H_q(\Phi^c\cap U,\Phi^c\cap U\backslash\{u\}:\mathbb{F})$.

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