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There are analogues of certain basic notions in algebraic topology in the theory of Banach spaces. For example, the Brouwer fixed point theorem generalizes to the Schauder fixed point theorem, and the idea of the degree of a map generalizes to the Leray-Schauder degree. In both cases, you must restrict yourself to considering (nonlinear) compact operators: operators that take bounded sets to relatively compact sets.

How far can this analogy be extended? How much of algebraic topology can be made to work in this setting? Suppose we consider arbitrary bounded subsets (not necessarily convex) of a Banach space as our class of spaces, and then compact operators between those. (If that's the wrong choice, feel free to correct it.) Can you define some sort of infinite-dimensional homology group so that the degree is an element of it? is there an analogue of the Lefschetz fixed point theorem? Or does the analogy break down if you try to go beyond defining a degree?

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    $\begingroup$ I think it is a mistake to view the Schauder fixed point theorem as in any way a generalization of the Banach fixed point theorem. First of all, the latter is really a purely metric space (not Banach space) theorem and is connected to the ideas of distance decreasing and completeness; it also does not depend in any way with the global topology of the space. The Schauder theorem on the other hand is a convexity related result, and the proof is in no way related to that of the Banach fixed point theorem. $\endgroup$ Commented Oct 27, 2010 at 17:19
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    $\begingroup$ He possibly means the Brouwer's fixed point theorem, rather than Banach's contraction principle. $\endgroup$ Commented Oct 27, 2010 at 17:24
  • $\begingroup$ I did mean Brouwer, not Banach. Sorry about that. $\endgroup$
    – arsmath
    Commented Oct 28, 2010 at 7:16
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    $\begingroup$ Ten years later, and I still regularly say Banach when I mean Brouwer, and vice versa. $\endgroup$
    – arsmath
    Commented Feb 11, 2021 at 9:13

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Critical groups capture more information than Leray Schauder degree. Instead of a number, critical groups are a sequence of homological groups, which depends on local behaviour in the variational setting. For more details, see the book infinite dimensional Morse theory by Kung Ching Chang.

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    $\begingroup$ Do you mean this book: doi.org/10.1007/978-1-4612-0385-8 ? $\endgroup$
    – David Roberts
    Commented Feb 11, 2021 at 9:13
  • $\begingroup$ @DavidRoberts Yes, that’s the book. It also occurred to me that Conley index is an extension of critical groups for isolated critical sets instead of points. Conley index is illustrated in the book methods in nonlinear analysis by the same author. $\endgroup$
    – Qi Sun
    Commented Feb 12, 2021 at 10:21

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