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Geometric realizations of the multiplihedra

Part of: Categories with structure Curves Polytopes and polyhedra

Published online by Cambridge University Press:  12 May 2010

S. Ma’u  and
C. Woodward
S. Ma’u
Affiliation:
The Mathematics Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720-5070, USA (email: sikimeti@msri.org)
C. Woodward
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA (email: ctw@math.rutgers.edu)
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Abstract

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We realize the multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We show that this moduli space is the non-negative real part of a complex moduli space of stable scaled marked curves.

Keywords

multiplihedronassociahedronnodal diskmetric treemoment polytope

MSC classification

Secondary: 18D50: Operads 14H10: Families, moduli (algebraic) 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 1002 - 1028
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Boardman, J. M. and Vogt, R. M., Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, vol. 347 (Springer, Berlin, 1973).CrossRefGoogle Scholar
[2]Forcey, S., Convex hull realizations of the multiplihedra, Topology Appl. 156 (2008), 326–347.CrossRefGoogle Scholar
[3]Fukaya, K. and Oh, Y.-G., Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1 (1997), 96–180.CrossRefGoogle Scholar
[4]Fukaya, K., Oh, Y.-G., Ohta, H. and Ono, K., Lagrangian intersection Floer theory: anomaly and obstruction. Parts I, II, AMS/IP Studies in Advanced Mathematics, vol. 46 (American Mathematical Society, Providence, RI, 2009).Google Scholar
[5]Fulton, W., Introduction to toric varieties, in The William H. Roever lectures in geometry, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
[6]Iwase, N. and Mimura, M., Higher homotopy associativity, in Algebraic topology (Arcata, CA, 1986), Lecture Notes in Mathematics, vol. 1370 (Springer, Berlin, 1989), 193–220.CrossRefGoogle Scholar
[7]McDuff, D. and Salamon, D., J-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52 (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
[8]Nguyen, K. and Woodward, C., Morphisms of cohomological field theories, Preprint (2009), arXiv:math.AG/0903.4459.Google Scholar
[9]Sottile, F., Toric ideals, real toric varieties, and the moment map, in Topics in algebraic geometry and geometric modeling, Contemporary Mathematics, vol. 334 (American Mathematical Society, Providence, RI, 2003), 225–240.CrossRefGoogle Scholar
[10]Stasheff, J. D., Homotopy associativity of H-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275–292; J. D. Stasheff, Homotopy associativity of H-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 293–312.Google Scholar
[11]Stasheff, J., H-spaces from a homotopy point of view, Lecture Notes in Mathematics, vol. 161 (Springer, Berlin, 1970).CrossRefGoogle Scholar
[12]Ziltener, F., Symplectic vortices on the complex plane and quantum cohomology, PhD thesis, Zurich, (2006).Google Scholar