Small covers and the equivariant bordism classification of 2-torus manifolds
Associated with the Davis–Januszkiewicz theory of small covers, this paper deals with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We define a differential operator on the “dual” algebra of the unoriented -representation algebra introduced by Conner and Floyd, where . With the help of -colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tom Dieck–Kosniowski–Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the -equivariant unoriented bordism classification of -dimensional 2-torus manifolds. We show that the -equivariant unoriented bordism class of each -dimensional 2-torus manifold contains an -dimensional small cover as its representative, solving the conjecture posed in . In addition, we also obtain that the graded noncommutative ring formed by the equivariant unoriented bordism classes of 2-torus manifolds of all possible dimensions is generated by the classes of all generalized real Bott manifolds (as special small covers over the products of simplices). This gives a strong connection between the computation of -equivariant bordism groups or ring and the Davis–Januszkiewicz theory of small covers. As a computational application, with the help of computer, we completely determine the structure of the group formed by equivariant bordism classes of all 4-dimensional 2-torus manifolds. Finally, we give some essential relationships among 2-torus manifolds, coloring polynomials, colored simple convex polytopes, colored graphs.
Key words and phrases:Small cover, tom Dieck–Kosniowski–Stong localization theorem, 2-torus manifold, bordism, colored graph.
2000 Mathematics Subject Classification:Primary 55N22, 57R85, 13A02, 57S17; Secondary 05C15, 52B11.
Throughout this paper, assume that is the mod 2-torus group of rank . An -dimensional 2-torus manifold is a smooth closed -dimensional (not necessarily oriented) manifold equipped with an effective smooth -action, so its fixed point set is empty or consists of a set of isolated points (see [19, 21]). The seminal work of Davis and Januszkiewicz in  discussed a special kind of 2-torus manifolds, so called small covers (as the topological versions of real toric varieties), each of which admits a locally standard -action such that the orbit space of action is homeomorphic to a simple convex polytope. Small covers provide a strong link between equivariant topology, polytope theory and combinatorics. This paper will deal with the theory of 2-torus manifolds from the viewpoint of equivariant bordism. We still expect such a link like small covers although 2-torus manifolds form a much wider class than small covers.
In 1960s, Conner and Floyd began with the study of the equivariant bordism theory of smooth closed -manifolds ([9, 10]). They introduced and studied a graded commutative algebra over with unit, , where consists of -equivariant unoriented bordism classes of all smooth closed -dimensional manifolds with smooth -actions fixing a finite set, and the addition and the multiplication on are defined by the disjoint union and the cartesian products with diagonal -actions of -manifolds, respectively. We note that has also a -linear space structure. As stated in [9, p.75] and [10, p.107], an action of on is equivalent to a collection of involutions , with , which means that is generated by . We also require the -action to be effective. Thus, consists of equivariant unoriented bordism classes of all -dimensional 2-torus manifolds, also denoted by in .
Conner and Floyd showed in [9, 10] that when , and when , where denotes the class of with the standard -action. When , the group structure of was determined in  (see also ), and it was also shown therein that . However, as far as authors know, when , the ring structure of is still open, and the group structure of with is also so.
With the above understood, now our purpose in this paper can be equivalently restated as follows: we shall concentrate on the study of as a group generated by equivariant unoriented bordism classes of all -dimensional 2-torus manifolds. Given a nonzero class in , we know by classical results of Stong  that the fixed point set of is a nonempty finite set, and the collection of the associated tangential -representations at the fixed points, which is considered as a squarefree homogenous polynomial of degree in , determines the class , where is the graded polynomial algebra over generated by all irreducible -representations. In other words, there is a monomorphism (see also Subsection 2.1). We shall consider the following questions:
What squarefree homogenous polynomials in arise as fixed point data of 2-torus manifolds?
Are there preferred representatives in the equivariant unoriented bordism classes of ? With respect to this question, the following concrete conjecture was posed in .
Conjecture (): Each class of contains a small cover as its representative.
On (Q1), there is a theoretical answer stemming from the tom Dieck–Kosniowski–Stong localization theorem in terms of an integrality condition for the tangential fixed point data, which applies to smooth closed -manifolds of an arbitrary dimension (not necessarily equal to ) with finite fixed point set. In the special case of (i.e., 2-torus manifolds), we shall formulate a simple criterion in terms of the vanishing of a differential on the dual of the given squarefree homogenous polynomial (see Theorem 2.3). This description is based upon the consideration of -colored graphs or mod 2 GKM graphs, which was introduced and studied in [2, 3, 18, 20]. Furthermore, we may use this simple criterion to consider (Q2). We show that each -dimensional 2-torus manifold is -equivariantly bordant to an -dimensional small cover, giving an affirmative answer to Conjecture () (see Theorem 2.5). As shown in , a small cover over a product of simplices is a generalized real Bott manifold. We also show that as a graded noncommutative ring, is generated by the classes of all generalized real Bott manifolds (see Theorem 2.6), where the multiplication on is defined by the cartesian product of actions (see Subsection 2.2 for details). This provides a strong link between the computation of -equivariant bordism groups or ring and the Davis–Januszkiewicz theory of small covers. As a computational application, we use a computer program to get that if , then the -linear space produced by equivariant bordism classes of all 4-dimensional 2-torus manifolds has dimension 510 (see Proposition 2.7).
This paper is organized as follows. We shall review the tom Dieck–Kosniowski–Stong localization theorem and give the statements of main results in Section 2. In Section 3 we introduce the notions of faithful polynomials and its dual polynomials, and give the definition of the differential operator on the free polynomial algebra . In Section 4 we review the basic theories of colored graphs and small covers. In particular, we also discuss the decomposability of -colored simple convex polytopes. In Section 5 we give the proof of Theorem 2.3. In Section 6 we introduce the linear spaces and , and then use them to study the structure of and to finish the proofs of Theorems 2.5–2.6. In Section 7 we give a summary on some essential relationships among 2-torus manifolds, coloring polynomials, colored simple convex polytopes, colored graphs, and also pose some problems. In Section 8, for a local completeness we give a simple proof to show how the tom Dieck–Kosniowski–Stong localization theorem follows from the existence theorem of tom Dieck. Finally we give an algorithm of determining a basis of in Section 9.
2. tom Dieck–Kosniowski–Stong localization theorem and statements of main results
2.1. tom Dieck–Kosniowski–Stong localization theorem
Following [9, 10], let be the linear space over , generated by the isomorphism classes of -dimensional real -representations. Then becomes a graded commutative algebra over with unit, called the Conner–Floyd unoriented -representation algebra here, where the multiplication is given by the direct sum of representations, and the unit is given by the representation of degree 0. As pointed out in , is not the Grothendieck ring of -representations, an entirely different concept. It is well-known that each irreducible real -representation is one-dimensional, so is also the graded polynomial algebra over generated by the isomorphism classes of one-dimensional irreducible real -representations. There is the following essential relation between and , due to Stong  (see also  and ).
Theorem 2.1 (Stong).
defined by is a monomorphism as algebras over where denotes the real -representation on the tangent space at .
Given a class in , as mentioned in Section 1, the action of on is effective. If the fixed point set of the -action on is empty, then in by a result of Stong  or [10, Theorem 31.2]. Thus, if is nonzero in , then since the action of on is effective. This implies that if , then would be trivial. It should be pointed out that in , the class of a single point with a -action is used as the unit in , which represents the generator of so . This enriches the algebraic structure of , and does not bring any essential influence on the study of although a -action on a single point is trivial. In this paper we use this convention in .
In , tom Dieck showed that the equivariant unoriented bordism class of a smooth closed -manifold is completely determined by its equivariant Stiefel–Whitney characteristic numbers, and in particular, the existence of a -manifold fixing a finite set can be characterized by the integral property of its fixed point data (see [12, Theorem 6]). Later on, Kosniowski and Stong  gave a more precise localization formula for the characteristic numbers of in terms of the fixed point data. Then the existence theorem of tom Dieck can be formulated into the following localization theorem in terms of Kosniowski and Stong’s localization formula (see [17, §5 of p.740]). Here for a completeness, we shall give a simple proof for how the following theorem follows from the existence theorem of tom Dieck in Section 8 of this paper.
Theorem 2.2 (tom Dieck–Kosniowski–Stong localization theorem).
Let be a collection of faithful -representations in . Then a necessary and sufficient condition that or is the fixed point data of a -manifold is that for all symmetric polynomial functions over ,
where denotes the equivariant Euler class of , which is a product of nonzero elements of , and means that variables in the function are replaced by those degree-one factors in .
Although all elements of can be characterized by the formula (2.1), it is still quite difficult to determine the algebra structure of . Also, in Theorem 2.2, if is the fixed data of a -manifold , then the polynomial (2.1) is exactly an equivariant Stiefel–Whitney number of . Actually, if we formally write the equivariant total Stiefel–Whitney class of the tangent bundle as , then the equivariant Stiefel–Whitney number can be calculated by the formula
2.2. Statements of main results
Now let (resp. ) denote the set of all homomorphisms (resp. ). Then both and have natural abelian group structures given by those of and in the usual way (i.e., the addition is given by ) and they have also linear space structures over . Let (resp. ) be the graded polynomial algebra over the linear space (resp. ), i.e., the infinite symmetric tensor algebra over (resp. ). Since both and are isomorphic to as linear spaces, we have that
On the other hand, it is well-known that all irreducible real -representations bijectively correspond to all elements in , where every irreducible real representation of has the form with for , and is trivial if for all . Thus, if by we denote the free polynomial algebra over , then can be identified with , where means the set obtained by forgetting the algebraic structure on . Similarly, we may define in the same way as . We note that is generated by elements of , while is generated by a basis (containing elements) of .
In a certain sense, both and are dual to each other. Thus, given a faithful -polynomial in (which means that for each monomial of , the set is a basis of ), we can obtain a unique dual -polynomial in (see also Subsection 3.1). In Subsection 3.2 we shall define a differential operator on . Identifying with , we may regard as a subring of . Then the following result gives another characterization of in terms of .
Let be a faithful -polynomial in . Then if and only if .
We shall see from Theorem 4.2 in Subsection 4.1 that can also be characterized by a -colored graphs (or mod 2 GKM graph), so that we may use the -colored graphs to give the proof of Theorem 2.3. On the other hand, there is also an essential relation between -colored graphs and -colored simple convex -polytopes in the setting of small covers, which indicates an algebraic duality (see Proposition 4.7). This is an important reason why we consider the dual polynomial of .
Let be a faithful -polynomial in . Then if and only if for all symmetric polynomial functions over ,
when and are regarded as polynomials in .
Our next task is to apply Theorem 2.3 to the study of .
Since is a monomorphism, it follows by Theorem 2.3 that as linear spaces over , is isomorphic to the linear space formed by all faithful -polynomials with . Then, the problem can be further reduced to studying the linear space formed by the dual polynomials of those polynomials in (see Section 6). Based upon this and the Davis–Januszkiewicz theory of small covers, we will show that
The Conjecture holds for arbitrary dimension .
The sum is also a graded ring with the multiplication defined by where the -action on is given by by regarding as . It should be pointed out that the multiplication defined as above depends upon the ordering of the cartesian product of with -action and with -action. Actually, in the same way as above, by regarding as , the -action on would be defined by . However, generally such two -actions on and are not equivariantly cobordant except for , but up to automorphisms of , they have not any difference essentially (i.e., by using an automorphism of , one of such two actions can be changed into the other one). Thus, is a graded noncommutative ring.
is generated by the classes of all small covers over with , where is an -simplex.
We know from  that a small cover over a product of simplices is actually identified with a generalized real Bott manifold, so is generated by the classes of all generalized real Bott manifolds.
As a computational application, we determine the precise structure of .
is generated by merely the classes of small covers over , and has dimension .
3. Faithful polynomials, dual polynomials and a differential operator
3.1. Faithful polynomials and dual polynomials
and are clearly isomorphic to , and they are dual to each other by the following pairing:
defined by , composition of homomorphisms. For example, the standard basis of gives the dual basis of , where is defined by , and is defined by .
Recall that can be identified with , such that each monomial in can be used as the class of a -representation in . Suppose that is a nonzero homogeneous polynomial of degree in such that each monomial is the class of an -dimensional faithful -representation, so forms a basis of . Such a polynomial is called a faithful -polynomial of . By the pairing (3.1), determines a dual basis of . Furthermore, we obtain a unique homogeneous polynomial in , which is called the dual -polynomial of .
When , take a faithful polynomial in . Then the dual polynomial of is in .
3.2. A differential operator on
We define a differential operator on as follows: for each monomial of degree
and , where the symbol means that is deleted. Obviously, . Thus, forms a chain complex.
For all , .
It is easy to see that . So it suffices to show that for . Obviously, . Conversely, for any , take where . Then so . Thus . ∎
A polynomial is said to be squarefree if each monomial of is a product of distinct nontrivial elements in , where the trivial element in is the zero homomorphism from to .
Let be squarefree. Then if and only if there is a squarefree polynomial in such that .
Obviously, if , then . Conversely, by Leibniz rule, for any nontrivial element , . If is not squarefree, then we may write such that is nonzero and sequarefree. Furthermore, . This forces to be zero since is squarefree. Thus, we can take as desired. ∎
It should be pointed out that similarly we may define a differential operator on . However, given a faithful polynomial , if , then generally we cannot obtain . For example, for the and in Example 3.1, a direct calculation shows that but .
Let . For an automorphism of , let denote the polynomial of produced by replacing each degree-one factor in by , where is regarded as an element in . Then we see that naturally admits an action of , defined by . A direct calculation gives the following result.
Let and . Then
4. -colored graphs and small covers
Throughout the following, will be identified with . Then we may write the Stong homomorphism as .
4.1. -colored graphs
In , Goresky, Kottwitz and MacPherson established the GKM theory, indicating that there is an essential link between topology and geometry of torus actions and the combinatorics of colored graphs (see also ). Such a link has already been expanded to the case of mod 2-torus actions (see, e.g., –, , and ). Specifically, assume that is a smooth closed -manifold with an effective smooth -action fixing a nonempty finite set , which implies (see ). Then we know from [18, 20] that the -action on defines a regular graph of valence with the vertex set and a -coloring .
In this paper we shall pay more attention on the extreme case in which (i.e., is a 2-torus manifold). In this case we also know from [2, 18, 20] that such a -colored graph is uniquely determined by the -action where is defined as a map from the set of all edges of to all non-trivial elements of , and it satisfies the following properties:.
for each vertex of , is faithful in (or equivalently, forms a basis of ), where denotes the set of all edges adjacent to ;
for each edge of , in where and are two endpoints of .
The pair is called the -colored graph of the 2-torus manifold here.
The property (P2) has the following equivalent statement that for each edge of , there is a unique bijection such that for any ,
The collection is called a connection of . Geometrically, or means the collection of all tangential -representations at fixed points in , and means the connection among all tangential -representations at fixed points.
Consider the -dimensional real projective space () with the standard linear -action defined by
which fixes isolated points with 1 in the -th place for . This action determines a unique regular graph , which is just the 1-skeleton of an -simplex , and the edges of are colored by , respectively, where is the standard basis of . When , the -colored graph is shown in Figure 1.
We note that the diagonal action on two copies of the standard -action on and the twist involution on the product may give a -action on fixing three fixed points. However, with this