Define a new SL(2,C) Casson invariant for 3-manifolds that incorporates information from the higher dimensional components of the character variety of the 3-manifold group. Explore the relationship between the newly defined SL(2,C) Casson invariant and the A-polynomial. The main di culty is non-compactness of the character variety, and so one needs to work with compactifications of the character variety. Various ways of compactifying have been introduced, and we intend to work with "wonderful" compactifications that preserve the algebraic structure of the character variety.
Impact
This project will give an alternate approach to understanding the Euler characteristic of the recently defined SL(2,C) Floer theory (see 2017 preprint of Abouzaid and Manolescu). The mathematical machinery developed will be useful in other applications of algebraic intersection theory and gauge theory, for instance in defining new invariants of knots and 3-manifolds using other non-compact Lie groups.
