Do there exist some relations between Functional Analysis and Algebraic Topology?

Solution 1:

Yes, there is a whole field of study called Non-Commutative Topology/Geometry that is centred around this idea. K-theory for C* algebras, Brown-Douglas-Filmore's study of essentially normal operators, Kasparov's work on the Novikov conjecture, etc. are all very beautiful.

Edit: Given your background, there are two possible ways I can think of to begin :

a) Start learning some Operator Algebras' theory (Gerard Murphy's book is good for this). Then graduate to K-theory for C* algebras (from Rordam/Laustsen/Larsen's book)

b) Start with Vector bundles and K-theory for topological spaces (Allen Hatcher has some notes on this). Then read Atiyah's manuscript on K-theory.

This is just my 2c.

Solution 2:

See Atiyah–Singer index theorem: http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem.

Solution 3:

An interesting fact is that the space of Fredholm operators on a Hilbert space classifies stable equivalence classes of vector bundles over any space: This is the topological $K$ theory of the space. This is the Atiyah-Jänich Theorem.