## Algebraic & Geometric Topology

### Quilted strips, graph associahedra, and $A_\infty$ $n$–modules

Sikimeti Ma’u

#### Abstract

We consider moduli spaces of quilted strips with markings. By identifying each compactified moduli space with the nonnegative real part of a projective toric variety, we conclude that it is homeomorphic under the moment map to the moment polytope. The moment polytopes in these cases belong to a certain class of graph associahedra, which include the associahedra and permutahedra as special cases. In fact, these graph associahedra are precisely the polytopes whose facet combinatorics encode the $A∞$ equations of $A∞$ $n$–modules.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 2 (2015), 783-799.

Dates
Received: 9 December 2013
Revised: 10 September 2014
Accepted: 8 January 2015
First available in Project Euclid: 28 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1511895789

Digital Object Identifier
doi:10.2140/agt.2015.15.783

Mathematical Reviews number (MathSciNet)
MR3342676

Zentralblatt MATH identifier
1327.14135

Subjects
Primary: 14H10: Families, moduli (algebraic)
Secondary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]

#### Citation

Ma’u, Sikimeti. Quilted strips, graph associahedra, and $A_\infty$ $n$–modules. Algebr. Geom. Topol. 15 (2015), no. 2, 783--799. doi:10.2140/agt.2015.15.783. https://projecteuclid.org/euclid.agt/1511895789

#### References

• J,M Bloom, A link surgery spectral sequence in monopole Floer homology, Adv. Math. 226 (2011) 3216–3281
• M,P Carr, S,L Devadoss, Coxeter complexes and graph-associahedra, Topology Appl. 153 (2006) 2155–2168
• M Davis, T Januszkiewicz, R Scott, Fundamental groups of blow-ups, Adv. Math. 177 (2003) 115–179
• S,L Devadoss, A realization of graph associahedra, Discrete Math. 309 (2009) 271–276
• W Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton Univ. Press (1993)
• D McDuff, D Salamon, $J\!$–holomorphic curves and symplectic topology, 2nd edition, AMS Colloq. Publ. 52, Amer. Math. Soc. (2012)
• A Postnikov, Permutohedra, associahedra, and beyond, Int. Math. Res. Not. 2009 (2009) 1026–1106
• F Sottile, Toric ideals, real toric varieties, and the moment map, from: “Topics in algebraic geometry and geometric modeling”, (R Goldman, R Krasauskas, editors), Contemp. Math. 334, Amer. Math. Soc. (2003) 225–240
• V Toledano Laredo, Quasi-Coxeter algebras, Dynkin diagram cohomology, and quantum Weyl groups, Int. Math. Res. Pap. IMRP (2008)
• K Wehrheim, C,T Woodward, Quilted Floer cohomology, Geom. Topol. 14 (2010) 833–902