Low-dim topology and algebraic geometry: P. Popov

Dear students,

Please join us at the next meeting of the Low-dimensional Topology and Algebraic Geometry seminar.

https://math.hse.ru/lowdimtop/

As usual, there will be plenty of snacks!

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Math Department of the Higher School of Economics,
Room 326
Thursday March 22, 10:30-12:00

Speaker: Pavel Popov

Title: Character varieties and Heegaard splittings (Part 2)

Abstract:

In the last talk we introduced a character variety X(Gamma) for agroup Gamma. It is a variety of representations of this group intoSL(2,C). We calculated the tangent spaces to regular irreduciblerepresentations.  If T is a topological space, we denote X(pi_1(T))just as X(T). In this talk we consider the case of a 3-manifold M witha Heegaard decomposition into U_1 and U_2 along a surface S. It turnsout that the smooth locus of X(S) has a symplectic structure andX(U_1), X(U_2) are Lagrangian subvarieties. Using intersection theoryfor X(U_1), X(U_2) we introduce an invariant [Curtis01] lambda(M) of3-manifolds M which, roughly speaking, counts isolated irreduciblerepresentations of pi_1(M). In the end we discuss some results aboutthis invariant and compare it with the more sophisticated invariant of[AM17] which is constructed using the "derived lagrangianintersection" of X(U_1) and X(U_2).

[Curtis01] Cynthia L.Curtis, An intersection theory count of theSL2(C)-representations of the fundamental group of a 3-manifold.

[AM17] Mohammed Abouzaid, Ciprian Manolescu, A sheaf-theoretic modelfor SL(2,C) Floer homology.