Algebraic topology for nonlinear compact operators - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T14:17:51Z http://mathoverflow.net/feeds/question/43830 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43830/algebraic-topology-for-nonlinear-compact-operators Algebraic topology for nonlinear compact operators arsmath 2010-10-27T16:30:37Z 2010-10-28T07:16:56Z <p>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 <a href="http://en.wikipedia.org/wiki/Schauder_fixed_point_theorem" rel="nofollow">Schauder fixed point theorem</a>, and the idea of the degree of a map generalizes to the <a href="http://en.wikipedia.org/wiki/Topological_degree_theory" rel="nofollow">Leray-Schauder degree</a>. In both cases, you must restrict yourself to considering (nonlinear) compact operators: operators that take bounded sets to relatively compact sets.</p> <p>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 <a href="http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem" rel="nofollow">Lefschetz fixed point theorem</a>? Or does the analogy break down if you try to go beyond defining a degree?</p>