#### Volume 11, issue 5 (2011)

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Sutured Floer homology, sutured TQFT and noncommutative QFT

### Daniel V Mathews

Algebraic & Geometric Topology 11 (2011) 2681–2739
##### Abstract

We define a “sutured topological quantum field theory”, motivated by the study of sutured Floer homology of product $3$–manifolds, and contact elements. We study a rich algebraic structure of suture elements in sutured TQFT, showing that it corresponds to contact elements in sutured Floer homology. We use this approach to make computations of contact elements in sutured Floer homology over $ℤ$ of sutured manifolds $\left({D}^{2}×{S}^{1},F×{S}^{1}\right)$ where $F$ is finite. This generalises previous results of the author over ${ℤ}_{2}$ coefficients. Our approach elaborates upon the quantum field theoretic aspects of sutured Floer homology, building a noncommutative Fock space, together with a bilinear form deriving from a certain combinatorial partial order; we show that the sutured TQFT of discs is isomorphic to this Fock space.

##### Keywords
TQFT, sutured Floer homology
##### Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 57R58, 57M27, 57R56
##### Publication
Received: 16 February 2011
Revised: 19 June 2011
Accepted: 23 June 2011
Published: 24 September 2011
##### Authors
 Daniel V Mathews Department of Mathematics Boston College Carney Hall, Room 301 Chestnut Hill MA 02467-3806 USA http://www.danielmathews.info