Asian Journal of Mathematics

Volume 22 (2018)

Number 1

Higher order generalization of Fukaya’s Morse homotopy invariant of $3$-manifolds, I: invariants of homology $3$-spheres

Pages: 111 – 180



Tadayuki Watanabe (Department of Mathematics, Shimane University, Matsue-shi, Shimane, Japan)


We give a generalization of Fukaya’s Morse homotopy theoretic approach for $2$-loop Chern–Simons perturbation theory to $3$-valent graphs with arbitrary number of loops at least $2$. We construct a sequence of invariants of integral homology $3$-spheres with values in a space of $3$-valent graphs (Jacobi diagrams or Feynman diagrams) by counting graphs in an integral homology $3$-sphere satisfying certain condition that is described by a set of ordinary differential equations.


Morse homotopy, Chern–Simons perturbation theory, homology $3$-sphere

2010 Mathematics Subject Classification

57M27, 57R57, 58D29, 58E05

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I would like to thank Professor Kenji Fukaya for encouraging me to write my result for publication. I would also like to thank Professor Masamichi Takase for valuable comments on spin 4-manifolds and would like to thank Professors Katrin Wehrheim and Tatsuro Shimizu for helpful comments. I am deeply grateful to the referees for the careful reading and for helpful comments. During the preparation and the revision of this paper, I have been supported by JSPS Grant-in-Aid for Young Scientists (B) 23740040 and 26800041.

Received 6 November 2015

Accepted 24 October 2016

Published 10 May 2018