Equivariant Cobordism Classification And Toric Toplogy

In this thesis, we focus on the equivariant cobordism classification theory, especially the equivariant cobordism classification of2-torus and torus manifolds with isolated fixed points. We gave a simple equivalent description of the tom Dieck-Kosniowski-Stong Lo-calization Theorem, determined the unoriented equivariant cobordism ring of2-torus man-ifolds and gave the generators of this ring. In particular, we also gave an answer to an open problem. In addition, we proved that a unitary toric manifold is determined by its equivariant Chern number given by the first and second chern classes. Associated to a conjecture posed by Kosniowski, we obtained a lower bound of the number of fixed points. At last, we proved that the map from the unitary toric equivariant cobordism group to2-torus equivariant cobordism group is an epimorphism. Our main results can be described in three parts as follows.In the first part, we study the equivariant cobordism of Zn2action. In general, the classification of equivariant cobordism is quite difficult. The structure of equivariant cobordism ring Z*of Zn2actions having isolated fixed points is also very hard to un-derstand. The most important result of this category is the torn Dieck-Kosniowski-Stong localization theorem, which provides the theoretical basis. In [26], Conner and Floyd de-termined the ring structure in the cases when n=1and n=2, but for n≥3there seems no way to solve the problem, and the ring structure still remains as a big riddle until now. The appearance of toric topology, especially the research on2-torus manifolds, bring us new ideas to solve these problems. An n-dimensional2-torus manifold is an n-dimensional close smooth manifold with an effective Zn2-action. For each2-torus manifold there is a2-torus coloring graph. This is a new relationship. All n-dimensional2-torus manifolds form a unoriented equivariant cobordism group under the disjoint union, denoted by mn, which is a main object of our research. Using the idea from coloring graphs, in [52] Prof. Zhi Lii determined the group m3and proved that the generators of this group can be chosen as small covers over simple polytopes, where the main tool used in that paper is the DKS localization theorem. Since the tom Dieck-Kosniowski-Stong localization theorem cannot work effectively in the high-dimensional cases, there are actually very big obstructions, so the problems become incredible when we consider the case n>3. On the other hand, in [52] Prof. Zhi Lii also gave a conjecture:Every class in mn contains a small cover as a generator. Enlighten by the dual relationship of coloring polytope and its coloring1skeleton, by constructing a new differential operation on the polynomial ring we gave a simple equivalent description of the tom Dieck-Kosniowski-Stong localization; which is our main innovation. Based upon this we can prove the conjecture mentioned above. We also proved that the base space polytopes of these small covers can be chosen as products of simplices. Then we determined the ring structure of9m*=(?)n≥mn. These are the main work of our Chapter three.In second part we studied the case of Tk-actions. In this case the main theoretical basis is the Atiyah-Bott-Berline-Vergne localization theorem, indicating that the values of equivariant cohomology classes on the fundamental class of a Tk-manifold can be de-scribed by the data of its fixed points. In [36], V. Guillemin, V. Ginzburg and Y. Karshon used this idea to study closed stable complex Tk manifolds M and their equivariant cobor-dism classification. Here a stable complex Tk-manifold M means that there is an effective Tk action on M and the tangent bundle of M admit a Tk stable complex structure. Here we also call this Tk-manifold a unitary Tk-manifold. Using Atiyah-Bott-Berline-Vergne localization theorem they proved that, if the Tk action on M contains only isolated fixed points, then Mn bounds equivariantly if and only if its equivariant chern number are all zero. In our case, we use the localization theorem to the2n-dimensional unitary Tn manifolds. This kind of manifolds can be considered as an extension of quasi-toric man-ifolds. In this case, using Atiyah-Bott-Berline-Vergne localization theorem, we proved that the equivariant cobordism class of a2n-dimensional unitory Tn manifold M is de-termined by the Chern numbers given by the first and second equivariant chern classes. In [45], Kosniowski conjectured that the lower bound of the number of the fixed points of an n-dimensional unitary S1-manifold which does not bounds equivariantly should be a linear function of the dimension n, and he also mentioned in a remark that the most likely function should be n/4+1. Using our result proved before, we can prove that if a2n-dimensional unitary Tn manifold doesn’t bound then the number of fixed points is at least [n/2]+1, where [n/2] denotes the minimal integer no less thann/2. These are the main work of our Chapter four.In the last chaper we studied the relationship of Zk2-actions and Tk-actions, so that this can deduce a lot of interesting problems. At first, we construct a conjugation map from the equivariant cobordism classes of2n-dimensional unitary Tn-manifolds to these of n dimensional Zn2-manifolds. Then, using the research results of our first two parts above, we proved that this map is an epimorphism. This means that in the sense of equivariant cobordism, the conjugation map from quasi-toric manifolds to small covers is surjective, too. These are the main results of this chapter. Associated to toric topology, we suggest some problems which should be worthy enough to further study at the end of this thesis.