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

The 2007 Abel Symposium came about on the college of Oslo in August 2007. The aim of the symposium used to be to collect mathematicians whose examine efforts have resulted in fresh advances in algebraic geometry, algebraic K-theory, algebraic topology, and mathematical physics. a standard subject matter of this symposium used to be the advance of latest views and new structures with a specific taste. 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 previous and new.

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For short we denote this by À . The same data also gives a topological crossed module ½ Ã . There is a homomorphism of crossed modules from À to ½ Ã , arising from this commuting square: Suppose that Ø ½ À Ø Ô Ã « Call this homomorphism . C. Baez and D. Stevenson ˇ Note that Ä ½ is just the ordinary Cech cohomology prove Lemma 2, we need to construct an inverse À ºÅ ½ Ã» ÀÄ ½ºÅ Ã». To ¬Ï ÀÄ ½ ºÅ Ã» ÀÄ ½ºÅ À » Let U Í be a good cover of Å ; then, as noted in Sect. 4 there is a bijection ÀÄ ½ºÅ Ã» ÀÄ ½ºU Ã» Hence to define the map ¬ it is sufficient to define a map ¬ Ï ÀÄ ½ ºU Ã» ˇ ÀÄ ½ºU À ».

This map between spheres was shown by Franks in [11] to be the relative attaching map in the Ï -structure of the complex M . º» º» ½ Ëº» º » º » the spheres u and Notice that in the special case when ½ have the same dimension ( ), and so the homotopy class of is given by its degree; by standard considerations this is the “count” #M of the number of points in the compact, oriented (framed) zero-manifold M (counted with sign). L. Cohen #M º » ½ º » (6) º » , extend to their The framings on the higher dimensional moduli spaces, M Æ compactifications, M , and they have the structure of compact, framed manifolds with corners.

Corollary. Let and both be simple. Then the function used to compactify Teichmuller space in Thurston’s work can be encoded in the algebraic structure of the bracket on with the given basis. Î To describe the next result note the following: If is a simple conjugacy class, for all integers and the bracket of the -th power of a with the -th power of a is zero, Ò Ñ . This is true because Ò and Ñ have disjoint representatives. Ò ¼ Ñ Ò Ñ Ë Theorem 2. [7]. Suppose is a surface with at least one puncture or boundary component and let be a conjugacy class which is not a power of another class.