By John C. Baez, Danny Stevenson (auth.), Nils Baas, Eric M. Friedlander, Björn Jahren, Paul Arne Østvær (eds.)

The 2007 Abel Symposium happened on the collage of Oslo in August 2007. The aim of the symposium was once to compile mathematicians whose examine efforts have resulted in contemporary advances in algebraic geometry, algebraic K-theory, algebraic topology, and mathematical physics. a typical subject matter of this symposium used to be the advance of latest views and new buildings with a express style. because the lectures on the symposium and the papers of this quantity reveal, those views and structures have enabled a broadening of vistas, a synergy among once-differentiated matters, and options to mathematical difficulties either outdated and new.

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22. S. Mac Lane and J. H. C. Whitehead, On the 3-type of a complex, Proc. Nat. Acad. Sci. 36 (1950), 41–48. 23. J. P. May, The Geometry of Iterated Loop Spaces, Lecture Notes in Mathematics 271, Springer, Berlin, 1972. 24. J. P. May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1 (1975), no. 155. 25. R. J. Milgram, The bar construction and abelian -spaces, Illinois J. Math. 11 (1967), 242–250. 26. J. Milnor, Construction of universal bundles II, Ann. Math. 63 (1956), 430–436. 27. I.

Given a commutative ring spectrum , when does the Floer complex £A ª £ have a realization by an -module spectrum as in Definition 1 in the introduction? Ultimately the criteria for such realizations rests on the properties of the flow category of the Floer functional, CA . This is defined as in the Morse theory case, where the objects are critical points and the morphisms are the spaces of piecewise Æ flow lines, M . The first criterion is that these moduli spaces can be given be given the structure of smooth, compact, manifolds with corners.

Any two extensions are homotopic because ¢ has contractible universal cover. Writing as ¼ union a two-disk we see any nontrivial embedded closed curve in can be isotoped into ¼ giving a nontrivial embedded curve distinct from the boundary. Conversely, if a nontrivial embedded curve on the surface with boundary is not freely homotopic to the boundary then it is still nontrivial in the closed surface. This proves the equivalence of £ and ££. 2 Two Properties of the Kernel Consider maps ¼ ¢ which are surjective on the fundamental group.