Integral Invariants of 3-Manifolds

TL;DR

This work constructs a hierarchy of configuration-space invariants for framed rational homology 3-spheres with trivial representation, connecting Axelrod–Singer and Kontsevich perspectives through a fixed Riemannian structure. Central to the approach is the explicit $A_\Theta$ invariant built from forms on the boundary and configuration spaces, complemented by a Chern–Simons correction, yielding a robust invariant $I_\Theta$. The framework extends to higher graph cocycles $\Gamma$, defining $I_\Gamma(M,f)=A_\Gamma(M)+\phi(\Gamma)\,\mathrm{CS}(M,f)$, and to knots in rational homology spheres via decorated knot graphs, producing knot invariants $I_\Gamma(K,M)=A_\Gamma(K,M)+\mu(\Gamma)\,\operatorname{sln}(K,M)$. Together, these results illuminate how graph-cohomology data translates into topological invariants via configuration-space integrals and Chern–Simons corrections, with explicit evaluations on $S^3$ and a clear pathway for higher-order invariants.

Abstract

This note describes an invariant of rational homology 3-spheres in terms of configuration space integrals which in some sense lies between the invariants of Axelrod and Singer and those of Kontsevich.

Figures (4)

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Theorems & Definitions (50)

• Lemma 2.1
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• Remark 3.3
• Proposition 3.4
• proof
• Remark 3.5
• Remark 3.6