Homotopy Quantum box conception (HQFT) is a department of Topological Quantum box thought based by way of E. Witten and M. Atiyah. It applies principles from theoretical physics to check crucial bundles over manifolds and, extra commonly, homotopy sessions of maps from manifolds to a set goal house. This booklet is the 1st systematic exposition of Homotopy Quantum box concept. It starts off with a proper definition of an HQFT and gives examples of HQFTs in all dimensions. the most physique of the textual content is targeted on $2$-dimensional and $3$-dimensional HQFTs. A research of those HQFTs results in new algebraic items: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. those notions and their connections with HQFTs are mentioned intimately. The textual content ends with a number of appendices together with an overview of contemporary advancements and an inventory of open difficulties. 3 appendices by means of M. Müger and A. Virelizier summarize their paintings during this region. The ebook is addressed to mathematicians, theoretical physicists, and graduate scholars drawn to topological features of quantum box thought. The exposition is self-contained and like minded for a one-semester graduate direction. necessities comprise in basic terms fundamentals of algebra and topology. A e-book of the eu Mathematical Society (EMS). disbursed in the Americas through the yankee Mathematical Society.

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G with g . ŒW / D zero has a boost to G zero , and the variety of such lifts is larger than or equivalent to j€j. the subsequent statement might be confirmed in part 7. 1. three. three Corollary. If W ¤ S 2 , then for any homomorphism g W D 1 . W / ! G, the quantity j Homg . ; G zero /j is divisible by means of j€j jZ. €/j2d 2 , the place d 1 is the genus of W and Z. €/ is the heart of €. If okay D C, then for all in Theorem 1. 2. 1, 2 H 2 . G I S 1 / H 2 . G I C /I (1. three. b) see part 2. for this reason g . /. ŒW / 2 S 1 and j g . /. ŒW / j D 1: (1. three. c) The formulation (1. 1. a), (1. 2. a), and (1. three. c) indicate the subsequent statement. 1. three. four Corollary. For any homomorphism g W D 1 . W / ! G, j Homg . ; G zero /j Ä j Hom. ; €/j: (1. three. d) This inequality is an equality if and provided that g. / G and g . /. ŒW / D 1 for all irreducible (finite-dimensional) complicated representations of €. The inequality (1. three. d) doesn't carry for arbitrary teams . for instance, allow G zero be the crowd of diversifications of the set f1; 2; 3g and permit q W G zero ! Z=2Z be the surjection wearing all transpositions to at least one . mod 2/. truly, € D Ker q D Z=3Z. enable be the two workforce with m 1 turbines x1 , : : : , xm and defining relatives x12 D x22 D D xm . A homomorphism ! € has to hold all of the turbines x1 , : : : , xm to an analogous aspect. consequently, j Hom. ; €/j D three. nonetheless, the homomorphism ! Z=2Z wearing x1 , : : : , xm to one . mod 2/ admits at the least 3m lifts to G zero sporting x1 , : : : , xm to arbitrary transpositions. a piece of a gaggle homomorphism p W zero ! is a homomorphism s W ! zero such that playstation D identification . The set of sections of p (possibly empty) is denoted S . p/. ninety two V Enumeration difficulties in measurement 1. three. five Corollary. permit p W zero ! D 1 . W / be a gaggle epimorphism with finite kernel ˆ. Then P jS . p/j D jˆj . jˆj= dim / . W / . ŒW /; 2Irr. ˆIK/ D the place . ŒW / 2 okay is the evaluate of 2 H 2 . I ok / on ŒW  2 H2 . I Z/. this can be received by way of surroundings in Theorem 1. 2. 1, G D and g D identification W ! . Corollary 1. three. four signifies that , G0 D zero , € D ˆ, q D p, jS . p/j Ä j Hom. ; ˆ/j: D and . ŒW / D 1 for all irreducible This inequality is an equality if and provided that advanced representations of ˆ. Corollary 1. three. three signifies that if W ¤ S 2 , then the quantity jS . p/j is divisible by means of jˆj jZ. ˆ/j2d 2 , the place d 1 is the genus of W . additional corollaries of Theorem 1. 2. 1 could be mentioned in Sections 7–9. V. 2 Linear representations of € and cohomology repair a illustration W € ! GLn . okay/ with n 1. We outline the cohomology category 2 H 2 . G I ok / showing in Theorem 1. 2. 1. We first introduce a cohomology classification Ä 2 H 2 . G I okay / that is slightly weaker than yet is outlined for all . Then, below an irreducibility-type assumption on , we outline . 2. 1 The cohomology category Ä . keep in mind the crowd G G from part 1. 2. for every ˛ 2 G , choose ˛Q 2 q 1 . ˛/ G zero . We suppose that 1Q D 1 2 G zero . For any ˛; ˇ 2 G , set  1 ˛Q ˇQ 2 € and h˛;ˇ D ˛ˇ IJ;ˇ D det . h˛;ˇ / 2 ok : the next lemma exhibits that fIJ;ˇ g˛;ˇ 2G (2. 1. a) is a normalized 2-cocycle on G . Its category in H 2 . G I okay / is denoted by way of Ä . it truly is transparent that Ä is an obstruction to the lifestyles of a homomorphism s W G !

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