# Topological methods in toric geometry, symplectic geometry and combinatorics (10w5026)

Arriving in Banff, Alberta Sunday, November 7 and departing Friday November 12, 2010

## Organizers

Tony Bahri (Rider University)

Fred Cohen (University of Rochester)

Matthias Franz (University of Western Ontario)

Samuel Gitler (Cinvestav, Sam Pedro Zacatenco)

Megumi Harada (McMaster University)

## Objectives

A short background for the subject matter is given next with objectives listed subsequently.

Spaces known now as generalized moment-angle complexes, (also called 'partial product spaces'), have a substantial and long history. H. Poincare and later, C.L. Siegel, initiated the study of dynamical systems on polytopes. More recently, S. Lopez de Medrano cite{Santiago} has made further contributions to this subject. In his 1960's thesis, G. Porter investigated the homotopy theory of special cases of these spaces and in the 1970's E.B. Vinberg developed a related topological construction cite{invitation,vinberg}. In the seminal work during the 1990s M. Davis and T. Januszkiewicz introduced a topological generalization of smooth, projective toric varieties which were being intensively studied by algebraic geometers cite{davis.jan}. They observed that every toric manifold is the quotient of a moment-angle complex by the free action of a real torus. Subsequently, a substantial body of excellent work on these inter-related subjects developed.

For topologists, algebraic geometers, and symplectic geometers,generalized moment-angle complexes have been studied in the contexts of toric manifolds and orbifolds, configuration spaces, complements of coordinate subspace arrangements, spaces of group homomorphisms and the moment map associated to a Hamiltonian action. There seems also to be a strong connection to the Cox homogeneous space associated to a toric variety.

Initially moment-angle complexes were used as a device which could identify various calculations in toric geometry as being purely topological in nature. An example is the difficult result that the equivariant cohomology of a smooth toric variety is given by the Stanley-Reisner ring of the fan. The computation becomes much more tractable when interpreted in the language of generalized moment-angle complexes. The ordinary integral cohomology of a smooth toric variety becomes easier from this point of view too.

The extent to which the topology of moment-angle complexes gives information about the underlying combinatorics continues to attract attention. Among the most important generalized moment angle complexes is the Davis-Januszkiewicz space of a simplicial complex. The cohomology of the Davis-Januszkiewicz space is the Stanley-Reisner ring of the simplicial complex. Examples of the significant current activity analyzing the homotopy properties of these spaces may be found among cite{bbcg}, cite{grbic.theriault}, cite{denham.suciu} and cite{notbohm.ray}.

Other applications to engineering and robotics have also emerged recently. For example, the space of motions of robot legs arising from "gait states" in hybrid controllers is being developed in joint work of D. Koditschek and one of the conference organizers. The starting point is the observation that certain moment-angle complexes can be identified precisely with the space of positions of robot legs as exemplified on the web by 'Rhex' for which the "gait states" are given by a moment-angle complex with underlying simplicial complex given by the $1$-skeleton of the $5$-simplex. Although these developments are still in a primitive state, moment-angle complexes naturally identify certain natural physical phenomena.

Currently, there is strong interest in the theory of singular toric varieties. The most accessible examples are toric orbifolds as their singularities are more controlled. The equivariant cohomology of a toric orbifold is the analogue of the Stanley-Reisner ring which arises as the equivariant cohomology of a toric manifold and of the Davis-Januskiewicz spaces. It has been shown recently cite{bfr} that the integral equivariant cohomology of certain toric orbifolds is given by piecewise polynomials on the underlying simplicial fan. For weighted projective spaces, this ring of piecewise polynomials has been computed. These have also been extensively studied from an equivariant symplectic point of view by M. Harada, T. Holm, R. Goldin, T. Kimura, and A. Knutson.

Given this compelling setting, some of the specific objectives of this conference are summarized as follows.

1. Identify the properties and structure of generalized moment-angle complexes which extend the specific known cases.

For example, the integral cohomology of ordinary moment-angle complexes and Davis-Januszkiewicz spaces has been important in toric topology as well as in algebraic geometry. The determination of the cohomology of the much more general construction follows as a very natural problem.

It is known that for any complex oriented theory, the cohomology ring of the Davis-Januszkiewicz spaces is the appropriate analogue of the Stanley-Reisner ring. Though much is known about the $KO$-theory of a toric manifold, an explicit description directly in terms of the characteristic data has not yet been given. Related questions include the behaviour of these spaces with respect to certain localization and completion operations.

It would be interesting to know whether there are implications of the stable splitting for moment-angle complexes for the moment map and the Kirwan surjectivity of symplectic geometry. (A moment-angle complex is the inverse image of a regular point under the moment map).

2. Develop the properties of recently discovered families of toric manifolds determined by a given one.

A new topological construction of a family of manifolds derived from a given one is not yet well understood from the point of view of algebraic geometry. The construction suggests that large classes of toric varieties may be specified by modified fan data which is combinatorially less complex than the original.

3. Moment-angle complexes and the complements of coordinate subspace arrangements.

It would be interesting to have a better understanding of the precise relationship among various quotient constructions in toric topology and toric geometry in the presence of singularities. It is known that every toric variety determined by a simplicial fan is the quotient of a Cox space by a group action. Characteristic function data provides a topological analogue by specifying a (not necessarily free) torus action on a moment-angle complex.

This family of topological spaces overlaps the class of toric varieties which have quotient singularities. These constructions are also related to complements of complex coordinate subspace arrangements. There is a wide interest in these, in particular in hyperplane complements.

One example is recent work of De Concini and Procesi cite{de.concini.procesi} as well as O.~Holtz and A.~Ron cite{holtz}. Yet another connection arises in the fundamental groups of these spaces where the Lie algebra attached to the descending central series is giving the unstable Bousfield-Kan spectral sequence for the homotopy groups of the two-sphere cite{CW}. It then becomes interesting to see whether there is geometric structure which impacts these.

4. Toric orbifolds, equivariant cohomology and $KO$-theory.

Approaches to this problem involving piecewise polynomials on the fan have linked together topologists, algebraic geometers and combinatorial theorists who study splines. Symplectic geometers are also very interested in this topic from the point of view of GKM theory. Of particular interest to them is the extent to which ideas of Kirwan surjectivity extend into the realm of equivariant cohomology.

It is known that integral piecewise polynomials defined on a fan may be represented in terms of ordinary homogeneous polynomials on a Euclidean space space lying "above" the fan. It would be very interesting to know whether this setting makes more tractable the ring of piecewise polynomials defined on the fan. Problems involving piecewise polynomials on fan are related to work on splines done by combinatorial theorists and algebraic geometers, L.~Billera, K.~Lee, C.~De Concini and C.~Procesi. The computation of the equivariant $KO$-theory of weighted projective spaces will complement nicely the recent computation of the equivariant cohomology, cite{bfr}.

A good description of the relationship the equivariant cohomology of toric orbifolds and their Chen-Ruan cohomology should prove to be useful. It is interesting that Chen-Ruan orbifold cohomology distinguishes among weighted projective spaces which are equivalent as toric varieties. Many of the same sorts of formulae involving the weights seem to appear in both calculations.

An important related question asks which toric varieties have a topological descriptions as quotient spaces? Recently M. Franz cite{franz} has extended the class of varieties for which this is true. An answer to this question is central to the computation of the equivariant cohomology.

Bibliography

{invitation} V. Buchstaber, and N. Ray, An Invitation to Toric Topology: Vertex Four of. a Remarkable Tetrahedron,February 2008.

{bbcg} A. Bahri, M. Bendersky, F.R. Cohen, and S. Gitler, The

polyhedral product functor, arXiv:0711.4689.

{bfr} A. Bahri, M. Franz and N. Ray, Piecewise Polynomials and the

Equivariant Cohomology of Weighted Projective Spaces. to appear

in the Mathematical Proceedings of the Cambridge Philosphical

Society.

{CW} F.R. Cohen, and J. Wu, Braid groups, free groups, and the loop space of the

2-sphere, math.AT/0409307.

{davis.jan} M. Davis, and T. Januszkiewicz, Convex polytopes,

Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no.

2, 417-451.

{de.concini.procesi}C. DeConcini, C. Procesi, The algebra of the box spline,

arXiv:math/0602019v1.

{denham.suciu} G. Denham and A. Suciu, Moment-angle complexes,

monomial ideals and Massey products, arXiv:math/0512497v4

[math.AT].

{franz} M. Franz, On the topology of toric varieties, in preparation.

{grbic.theriault} J. Grbic and S. Theriault, Homotopy type of the complement of a coordinate subspace arrangement

of codimension two, Russian Math. Surveys 59 (2004), no. 3,

1207-1209.

{hochster} M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial

complexes, in: Ring theory, II (Proc. Second Conf., Univ. Oklahoma,

Norman, Okla., 1975), pp. 171-223, Lecture Notes in Pure and Appl.

Math., vol. 26, Dekker, New York, 1977.

{Lisa.Megumi.Paul} M. Harada, L. Jeffrey, and P. Selick, private communication.

{holtz}O. Holtz, and A. Ron, Zonotopal algebra, arXiv:0708.2632v2

[math.AC].

{notbohm.ray} D. Notbohm, and N. Ray, On Davis-Januszkiewicz homotopy types. I. Formality and rationalization, AGT

textbf{5}(2005), 31-51.

{Santiago} Santiago Lopez de Medrano, Topology of the intersection of quadrics in $mathbb R^n$, in 'Algebraic Topology'

(Arcata Ca), Springer Verlag LNM textbf{1370}(1989), Springer

Verlag.

{vinberg} E.B. Vinberg, Discrete linear groups that are generated by

reflections (Russian), Izvestia: Math. 35(1971), 1072-1112.

Spaces known now as generalized moment-angle complexes, (also called 'partial product spaces'), have a substantial and long history. H. Poincare and later, C.L. Siegel, initiated the study of dynamical systems on polytopes. More recently, S. Lopez de Medrano cite{Santiago} has made further contributions to this subject. In his 1960's thesis, G. Porter investigated the homotopy theory of special cases of these spaces and in the 1970's E.B. Vinberg developed a related topological construction cite{invitation,vinberg}. In the seminal work during the 1990s M. Davis and T. Januszkiewicz introduced a topological generalization of smooth, projective toric varieties which were being intensively studied by algebraic geometers cite{davis.jan}. They observed that every toric manifold is the quotient of a moment-angle complex by the free action of a real torus. Subsequently, a substantial body of excellent work on these inter-related subjects developed.

For topologists, algebraic geometers, and symplectic geometers,generalized moment-angle complexes have been studied in the contexts of toric manifolds and orbifolds, configuration spaces, complements of coordinate subspace arrangements, spaces of group homomorphisms and the moment map associated to a Hamiltonian action. There seems also to be a strong connection to the Cox homogeneous space associated to a toric variety.

Initially moment-angle complexes were used as a device which could identify various calculations in toric geometry as being purely topological in nature. An example is the difficult result that the equivariant cohomology of a smooth toric variety is given by the Stanley-Reisner ring of the fan. The computation becomes much more tractable when interpreted in the language of generalized moment-angle complexes. The ordinary integral cohomology of a smooth toric variety becomes easier from this point of view too.

The extent to which the topology of moment-angle complexes gives information about the underlying combinatorics continues to attract attention. Among the most important generalized moment angle complexes is the Davis-Januszkiewicz space of a simplicial complex. The cohomology of the Davis-Januszkiewicz space is the Stanley-Reisner ring of the simplicial complex. Examples of the significant current activity analyzing the homotopy properties of these spaces may be found among cite{bbcg}, cite{grbic.theriault}, cite{denham.suciu} and cite{notbohm.ray}.

Other applications to engineering and robotics have also emerged recently. For example, the space of motions of robot legs arising from "gait states" in hybrid controllers is being developed in joint work of D. Koditschek and one of the conference organizers. The starting point is the observation that certain moment-angle complexes can be identified precisely with the space of positions of robot legs as exemplified on the web by 'Rhex' for which the "gait states" are given by a moment-angle complex with underlying simplicial complex given by the $1$-skeleton of the $5$-simplex. Although these developments are still in a primitive state, moment-angle complexes naturally identify certain natural physical phenomena.

Currently, there is strong interest in the theory of singular toric varieties. The most accessible examples are toric orbifolds as their singularities are more controlled. The equivariant cohomology of a toric orbifold is the analogue of the Stanley-Reisner ring which arises as the equivariant cohomology of a toric manifold and of the Davis-Januskiewicz spaces. It has been shown recently cite{bfr} that the integral equivariant cohomology of certain toric orbifolds is given by piecewise polynomials on the underlying simplicial fan. For weighted projective spaces, this ring of piecewise polynomials has been computed. These have also been extensively studied from an equivariant symplectic point of view by M. Harada, T. Holm, R. Goldin, T. Kimura, and A. Knutson.

Given this compelling setting, some of the specific objectives of this conference are summarized as follows.

1. Identify the properties and structure of generalized moment-angle complexes which extend the specific known cases.

For example, the integral cohomology of ordinary moment-angle complexes and Davis-Januszkiewicz spaces has been important in toric topology as well as in algebraic geometry. The determination of the cohomology of the much more general construction follows as a very natural problem.

It is known that for any complex oriented theory, the cohomology ring of the Davis-Januszkiewicz spaces is the appropriate analogue of the Stanley-Reisner ring. Though much is known about the $KO$-theory of a toric manifold, an explicit description directly in terms of the characteristic data has not yet been given. Related questions include the behaviour of these spaces with respect to certain localization and completion operations.

It would be interesting to know whether there are implications of the stable splitting for moment-angle complexes for the moment map and the Kirwan surjectivity of symplectic geometry. (A moment-angle complex is the inverse image of a regular point under the moment map).

2. Develop the properties of recently discovered families of toric manifolds determined by a given one.

A new topological construction of a family of manifolds derived from a given one is not yet well understood from the point of view of algebraic geometry. The construction suggests that large classes of toric varieties may be specified by modified fan data which is combinatorially less complex than the original.

3. Moment-angle complexes and the complements of coordinate subspace arrangements.

It would be interesting to have a better understanding of the precise relationship among various quotient constructions in toric topology and toric geometry in the presence of singularities. It is known that every toric variety determined by a simplicial fan is the quotient of a Cox space by a group action. Characteristic function data provides a topological analogue by specifying a (not necessarily free) torus action on a moment-angle complex.

This family of topological spaces overlaps the class of toric varieties which have quotient singularities. These constructions are also related to complements of complex coordinate subspace arrangements. There is a wide interest in these, in particular in hyperplane complements.

One example is recent work of De Concini and Procesi cite{de.concini.procesi} as well as O.~Holtz and A.~Ron cite{holtz}. Yet another connection arises in the fundamental groups of these spaces where the Lie algebra attached to the descending central series is giving the unstable Bousfield-Kan spectral sequence for the homotopy groups of the two-sphere cite{CW}. It then becomes interesting to see whether there is geometric structure which impacts these.

4. Toric orbifolds, equivariant cohomology and $KO$-theory.

Approaches to this problem involving piecewise polynomials on the fan have linked together topologists, algebraic geometers and combinatorial theorists who study splines. Symplectic geometers are also very interested in this topic from the point of view of GKM theory. Of particular interest to them is the extent to which ideas of Kirwan surjectivity extend into the realm of equivariant cohomology.

It is known that integral piecewise polynomials defined on a fan may be represented in terms of ordinary homogeneous polynomials on a Euclidean space space lying "above" the fan. It would be very interesting to know whether this setting makes more tractable the ring of piecewise polynomials defined on the fan. Problems involving piecewise polynomials on fan are related to work on splines done by combinatorial theorists and algebraic geometers, L.~Billera, K.~Lee, C.~De Concini and C.~Procesi. The computation of the equivariant $KO$-theory of weighted projective spaces will complement nicely the recent computation of the equivariant cohomology, cite{bfr}.

A good description of the relationship the equivariant cohomology of toric orbifolds and their Chen-Ruan cohomology should prove to be useful. It is interesting that Chen-Ruan orbifold cohomology distinguishes among weighted projective spaces which are equivalent as toric varieties. Many of the same sorts of formulae involving the weights seem to appear in both calculations.

An important related question asks which toric varieties have a topological descriptions as quotient spaces? Recently M. Franz cite{franz} has extended the class of varieties for which this is true. An answer to this question is central to the computation of the equivariant cohomology.

Bibliography

{invitation} V. Buchstaber, and N. Ray, An Invitation to Toric Topology: Vertex Four of. a Remarkable Tetrahedron,February 2008.

{bbcg} A. Bahri, M. Bendersky, F.R. Cohen, and S. Gitler, The

polyhedral product functor, arXiv:0711.4689.

{bfr} A. Bahri, M. Franz and N. Ray, Piecewise Polynomials and the

Equivariant Cohomology of Weighted Projective Spaces. to appear

in the Mathematical Proceedings of the Cambridge Philosphical

Society.

{CW} F.R. Cohen, and J. Wu, Braid groups, free groups, and the loop space of the

2-sphere, math.AT/0409307.

{davis.jan} M. Davis, and T. Januszkiewicz, Convex polytopes,

Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no.

2, 417-451.

{de.concini.procesi}C. DeConcini, C. Procesi, The algebra of the box spline,

arXiv:math/0602019v1.

{denham.suciu} G. Denham and A. Suciu, Moment-angle complexes,

monomial ideals and Massey products, arXiv:math/0512497v4

[math.AT].

{franz} M. Franz, On the topology of toric varieties, in preparation.

{grbic.theriault} J. Grbic and S. Theriault, Homotopy type of the complement of a coordinate subspace arrangement

of codimension two, Russian Math. Surveys 59 (2004), no. 3,

1207-1209.

{hochster} M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial

complexes, in: Ring theory, II (Proc. Second Conf., Univ. Oklahoma,

Norman, Okla., 1975), pp. 171-223, Lecture Notes in Pure and Appl.

Math., vol. 26, Dekker, New York, 1977.

{Lisa.Megumi.Paul} M. Harada, L. Jeffrey, and P. Selick, private communication.

{holtz}O. Holtz, and A. Ron, Zonotopal algebra, arXiv:0708.2632v2

[math.AC].

{notbohm.ray} D. Notbohm, and N. Ray, On Davis-Januszkiewicz homotopy types. I. Formality and rationalization, AGT

textbf{5}(2005), 31-51.

{Santiago} Santiago Lopez de Medrano, Topology of the intersection of quadrics in $mathbb R^n$, in 'Algebraic Topology'

(Arcata Ca), Springer Verlag LNM textbf{1370}(1989), Springer

Verlag.

{vinberg} E.B. Vinberg, Discrete linear groups that are generated by

reflections (Russian), Izvestia: Math. 35(1971), 1072-1112.