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 college of Oslo in August 2007. The target of the symposium used to be to assemble mathematicians whose study efforts have resulted in fresh advances in algebraic geometry, algebraic K-theory, algebraic topology, and mathematical physics. a standard topic of this symposium was once the advance of latest views and new structures with a specific style. because the lectures on the symposium and the papers of this quantity display, those views and structures have enabled a broadening of vistas, a synergy among once-differentiated topics, and strategies to mathematical difficulties either outdated and new.

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GT] 8. Chas M, Krongold F (work in progress) 9. Chas M (2004) Combinatorial Lie bialgebras of curves on surfaces. Topology 43:543–568. GT] 10. Chas M, Sullivan D (2004) Closed operators in topology leading to Lie bialgebras and higher string algebra. The legacy of Niels Henrik Abel, pp 771–784. GT] 11. Sullivan D Open and closed string topology. In: Graeme Segal 60th Proceedings 12. Abbaspour H (2005) On string topology of 3-manifolds. Topology 44(5):1059–1091. arXiv:0310112 13. Etingof P (2006) Casimirs of the Goldman Lie algebra of a closed surface.

B . Âº ½» Ë £ ½ ¼ 2. On morphisms of the form , the functor Ï induces the boundary homomorphism on homotopy groups: B ½ Ï Ï ª ª Ï B £ Proof. The proof mirrors the arguments given in [5] and [4] which concern realizing chain complexes of abelian groups by stable homotopy types. We therefore present a sketch of the constructions and indicate the modifications needed to prove this theorem. ÑÓ First suppose that Ï J¼ is a functor satisfying the properties (1) and (2) as stated in the theorem. The geometric realization has a natural filtration by -module spectra, ¼ ¸ ½ ¸ ¡¡¡ ¸ ½ ¸ ¸ ¡¡¡ where the th-filtration, , is the homotopy coequalizer of the maps and as in Definition 5, but where the wedges only involve the first -terms.

6 Proof of Theorem 2 The following proof was first described to us by Matt Ando (personal communication), and later discussed by Greg Ginot [15]. Suppose that is a simply-connected, compact, simple Lie group. Then the string group Ç of fits into a short exact sequence of topological groups ½ ÃºZ ¾» Ç ½ Ãº ¾» for some realization of the Eilenberg–Mac Lane space Z as a topological group. Applying the classifying space functor to this short exact sequence gives rise to a fibration Ç Ô Z Ãº ¿» We want to compute the rational cohomology of Ç .