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Torus actions and manifolds defined by simple polytopes
by
Taras E. Panov
Moscow State University
In our research we develop the study of relationship between the algebraic topology of manifolds and the combinatorics of polytopes. Originally, this research was inspired by the remarkable advances in both subjects due to the theory of toric varieties. The main object of our present study is the smooth manifold defined by the combinatorial structure of simple polytope. This manifold is equipped with natural action of the compact torus Tm. The study of such manifolds was undertaken in our joint paper with V. M. Buchstaber [BP1] and then developed in [BP2].
A convex n-dimensional polytope is called simple if there exactly n codimension-one faces meet at each vertex. We associate to each simple polytope Pn with m codimension-one faces a smooth (m+n)-dimensional manifold ZP with the canonical action of the compact torus Tm. A number of manifolds playing an important role in the different aspects of topology, algebraic and symplectic geometry could be obtained as the above manifolds ZP, or as quotients ZP/Tk for some toric subgroup Tk subset Tm that acts on ZP freely. It turns out that the maximal rank of the torus subgroup that can act on ZP freely equals m-n. We call quotients of ZP by tori of maximal possible rank m-n quasitoric manifolds. We use such a name because the important class of algebraic varieties known to algebraic geometers as toric manifolds fit the above picture. More precisely, one can use the above construction (i.e., the quotient of ZP by the torus subgroup) to obtain all smooth projective toric varieties (cf. [Da]). Originally, the quasitoric manifolds (under the name ``toric manifolds") appeared in [DJ].
Our approach to constructing manifolds defined by simple polytopes is based on one construction from the geometry of toric varieties. Namely, the one of the ways to define a toric manifold over a simple polytope Pn is to start from the affine algebraic set U(Pn) subset Cm with action of the algebraic torus (C*)m that is defined as the complement to a certain collection of coordinate planes in Cm defined by the combinatorics of Pn. The toric manifolds appear as quotients when one can find a subgroup D subset (C*)m isomorphic to (C*)m-n that acts on U(Pn) freely. The crucial fact in our approach is that it is always possible to find a subgroup R subset (C*)m isomorphic to (R+*)m-n, acting freely on U(Pn), and with quotient being a smooth manifold of dimension m+n. We refer to this manifold as defined by the initial simple polytope Pn. We also fix the action of the torus Tm on this manifold that is defined by the standard inclusion Tm subset Cm. The alternative approach to constructing manifolds defined by simple polytopes was proposed in [DJ], where these manifolds where defined as the quotient spaces ZP =Tm×Pn/ ~ for some natural equivalence relation ~ . We construct the equivariant embedding ie of this manifold into U(Pn) subset Cm and show that for the above subgroup R the composition ZP --> U(Pn) --> U(Pn)/R of embedding and orbit map is a diffeomorphism. Hence, from the topological viewpoint, both approaches give the same manifold, which we call the manifold defined by simple polytope Pn and denote ZP.
One of our main goals is to study the relationship between the combinatorial structure of simple polytopes and topology of the above described manifolds defined by these polytopes. There is a well-known algebraic invariant of a simple polytope - a special graded ring k(P) (here k is any field) called the face ring. This is the quotient ring of the polynomial ring k[v1, ... , vm] by a certain homogeneous ideal. Then one could consider the corresponding cohomology modules
The algebraic topology possesses two very powerful tools for studying the actions of compact groups on manifolds: the cobordism theory (including the theory of formal groups) and the index theory (the Atiyah-Singer theorem and the generalized Lefschetz formula). Both this methods were used by many authors for studying actions of different groups. The relations between the toric varieties and the cobordism theory were studied in [BR]. In our situation, both (quasi)toric manifolds and above described manifolds defined by polytopes produce a vast number of examples of torus actions. By use of the above general methods we show that many topological and cobordism invariants of these manifolds (such as the signature, the Todd genus etc.) can be expressed in purely combinatorial terms of the underlying simple polytope Pn.
Moreover, since the homotopy equivalence ZP =~ U(Pn) holds, our calculation of cohomology is also applicable to the set U(Pn). As it was mentioned above, the set U(Pn) is the complement to a certain collection of affine planes in Cm defined by the combinatorics of Pn. Hence, here we have a special case of the well-known general problem of calculation the cohomology of the complement to a collection of affine planes. In [GM, part III] there was proved the theorem that reduces this calculation to the calculation of the cohomology of a certain simplicial complex. In fact, our calculations show how the special properties of the collection of affine planes allow to obtain much more explicit description of the corresponding cohomologies, and moreover, calculate their multiplicative structure.
| [BP1] |
| [BP2] |
| [BR] |
| [Da] |
| [DJ] |
| [GM] |
| [St] |
Date received: June 2, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-29.