This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2005 session of the KSU REU "Brainstorming and Barnstorming". The main results are a decomposition theorem for quandles in terms of an operation of "semidisjoint union" showing that all finite quandles canonically decompose via iterated semidisjoint unions into connected subquandles, and a structure theorem for finite connected quandles with prescribed inner automorphism group. The latter theorem suggests a new approach to the classification of finite connected quandles.
Each Morita-Mumford-Miller (MMM) class e n assigns to each genus g ≥ 2 surface bundleWe prove that when n is odd the number e # n (E → M ) depends only on the diffeomorphism type of E, not on g, M or the map E → M . More generally, we prove that e # n (E → M ) depends only on the cobordism class of E. Recent work of Hatcher implies that this stronger statement is false when n is even. If E → M is a holomorphic fibering of complex manifolds, we show that for every n the number e # n (E → M ) only depends on the complex cobordism type of E.We give a general procedure to construct manifolds fibering as surface bundles in multiple ways, providing infinitely many examples to which our theorems apply. As an application of our results we give a new proof of the rational case of a recent theorem of Giansiracusa-Tillmann [GT, Theorem A] that the odd MMM classes e 2i−1 vanish for any surface bundle which bounds a handlebody bundle. We show how the MMM classes can be seen as obstructions to low-genus fiberings. Finally, we discuss a number of open questions that arise from this work.
We classify up to equivalence all finite-dimensional irreducible representations of PSL 2 (Z) whose restriction to the commutator subgroup is diagonalizable.
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