컨텐츠 시작
학술대회/행사
초록검색
| 제출번호(No.) | 0305 |
|---|---|
| 분류(Section) | Invited Talk |
| 분과(Session) | Topology (SS-07) |
| 영문제목 (Title(Eng.)) |
Equivariant cobordism classification and Davis--Januszkiewicz theory |
| 저자(Author(s)) |
Zhi L\"u1, Qiangbo Tan1 Fudan University1 |
| 초록본문(Abstract) | An $n$-dimensional {\em 2-torus manifold} is a smooth closed $n$-dimensional (not necessarily oriented) manifold equipped with an effective smooth $G$-action, where $G=(\mathbb Z_2)^n$. Associated with the Davis--Januszkiewicz theory of small covers, we deal with the theory of 2-torus manifolds from the viewpoint of equivariant cobordism. We define a differential operator on the ``dual" algebra of the unoriented $G$-representation algebra introduced by Conner and Floyd. With the help of $G$-colored graphs (or mod 2 GKM graphs), we may use this differential operator to give a very simple description of tomDieck--Kosniowski--Stong localization theorem in the setting of 2-torus manifolds. We then apply this to study the $G$-equivariant unoriented cobordism classification of $n$-dimensional 2-torus manifolds. We show that the $G$-equivariant unoriented cobordism class of each $n$-dimensional 2-torus manifold contains an $n$-dimensional small cover as its representative, giving an affirmative answer to the conjecture posed by the presenting author. In addition, we also obtain that the graded noncommutative ring formed by the equivariant unoriented cobordism 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 $G$-equivariant cobordism 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. |
| 분류기호 (MSC number(s)) |
55N22, 57R85, 13A02, 57S17, 05C15, 52B11 |
| 키워드(Keyword(s)) | 2-torus manifold, equivariant cobordism, small cover, localization theorem |
| 강연 형태 (Language of Session (Talk)) |
English |