By E. M. Friedlander, M. R. Stein
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The two decades because the booklet of this e-book were an period of constant progress and improvement within the box of algebraic topology. New generations of younger mathematicians were educated, and classical difficulties were solved, really during the software of geometry and knot conception.
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This exact quantity, as a result of a convention on the Chern Institute of arithmetic devoted to the reminiscence of Xiao-Song Lin, offers a extensive connection among topology and physics as exemplified via the connection among low-dimensional topology and quantum box conception. the amount comprises works on photo (2+1) - TQFTs and their functions to quantum computing, categorification and Khovanov homology, Gromov - Witten variety invariants, twisted Alexander polynomials, Faddeev knots, generalized Ricci circulate, Calabi - Yau difficulties for CR manifolds, Milnor's conjecture on quantity of simplexes, Heegaard genera of 3-manifolds, and the (A,B) - slice challenge.
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The following is a geometric example. 5 (Picard groups of algebras of functions) Let A = C ∞ (M ) be the algebra of smooth complex-valued functions on a manifold M . Using the Serre-Swan identiﬁcation of smooth complex vector bundles over M with projective modules over A, one can check that SPic(A) coincides with Pic(M ), the group of isomorphism classes of complex line bundles on M , which is isomorphic to H 2 (M, Z) via the Chern class map. We then have a purely geometric description of Pic(A) as Pic(C ∞ (M )) = Diﬀ(M ) H 2 (M, Z), (37) where the action of Diﬀ(M ) on H 2 (M, Z) is given by pull back.
This path lifting property suggests that complete Poisson maps play the role of “coverings” in Poisson geometry. This idea is borne out by some of the examples below. 23 (Complete functions) Let us regard R as a Poisson manifold, equipped with the zero Poisson bracket. ) Then any map f : P → R is a Poisson map, which is complete if and only if Xf is a complete vector ﬁeld. Observe that the notion of completeness singles out the subset of C ∞ (P ) consisting of complete functions, which is preserved under complete Poisson maps.
15 holds for n = 2 , but not in general. In fact, for n = 3, one can ﬁnd Π and Π , not in the same SO(n, n|Z)orbit, for which AΠ and AΠ are isomorphic (hence Morita equivalent) . 15 and its converse hold with respect to a reﬁned notion of Morita equivalence, called “complete Morita equivalence” , in which bimodules carry connections of constant curvature. For the algebraic Morita equivalence of smooth quantum tori, see . 15 was proven under an additional hypothesis. Rieﬀel and Schwarz consider three types of generators of SO(n, n|Z), and prove that their action preserves Morita equivalence.
Algebraic K-Theory by E. M. Friedlander, M. R. Stein