On representation varieties of 3-manifold groups

TL;DR

This paper proves universality theorems for the local structure of representation schemes of 3-manifold groups, showing they are as complex as schemes over rational numbers, revealing deep connections between topology and algebraic geometry.

Contribution

It establishes Murphy's Laws for representation schemes of 3-manifold groups, demonstrating their universality and complexity.

Findings

Germs of SL(2,C)-representation schemes are as complex as schemes over rationals.

Universality theorems apply to fundamental groups of closed 3-manifolds.

Representation schemes exhibit maximal algebraic complexity.

Abstract

We prove universality theorems ("Murphy's Laws") for representation schemes of fundamental groups of closed 3-dimensional manifolds. We show that germs of SL(2,C)-representation schemes of such groups are essentially the same as germs of schemes of over rational numbers.