Geometry, Topology and Categorification Seminar: Trisection invariants of 4-manifolds from Hopf algebras

Monday, April 12 at 2:30pm to 3:45pm

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Virtual Event


Shawn Cui, Purdue University

Abstract: The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras.  Here we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds.  In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant.  As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants.

Dial-In Information

Zoom meeting: https://usc.zoom.us/j/99698082212?pwd=eVVjNTcrSm5XL1Zaa2VGUTkvUkNtUT09

Meeting ID: 996 9808 2212
Passcode: 102110

Event Type

Lecture / Talk / Workshop

Department
Mathematics
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