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Homotopy theory, homological algebra, algebraic treatments of manifolds.
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A relative Kuiper theorem
Let $(H_0, \langle \,,\,\rangle_0)$ be a real separable Hilbert space,
and let $(H_1, \langle \,,\,\rangle_1)$ be a Hilbert space such that $H_1 \subset H_0$ is dense and such
that the inclusion $( …