Topic
Stable normal bundle
About: Stable normal bundle is a research topic. Over the lifetime, 52 publications have been published within this topic receiving 606 citations.
Papers published on a yearly basis
Papers
TL;DR: The derived cobordism theory of as discussed by the authors is a simplicial category for manifolds with stable normal bundles and can be imbedded into Euclidean space. But it is not a complete theory of manifolds and it cannot be used to define fundamental classes for all derived manifolds.
Abstract: We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local $C^\infty$-rings that is obtained by patching together homotopy zero-sets of smooth functions on Euclidean spaces.
We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection $A\cap B$ of submanifolds $A,B\subset X$ exists on the categorical level in our theory, and a cup product formula $$[A]\smile[B]=[A\cap B]$$ holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a {\em categorification} of intersection theory.
69 citations
TL;DR: The derived cobordism as mentioned in this paper is a simplicial category of derived manifolds, which contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold.
Abstract: We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C∞-rings that is obtained by patching together homotopy zero sets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection A∩B of submanifolds A,B⊂X exists on the categorical level in our theory, and a cup product formula [A]⌣[B]=[A∩B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.
45 citations
TL;DR: In this article, it was shown that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a Holomorphic fiber bundle.
Abstract: In this note we show that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a holomorphic fiber bundle. In the algebraic case, the fibration becomes trivial after a finite base change.
18 citations
TL;DR: A complete classification of simply connected 5-dimensional Poincare duality spaces up to oriented homotopy type is given in this paper, where the most important step is a method for describing the Spivak normal fibration and hence the exotic characteristic class.
Abstract: We give a complete classification of simply connected 5-dimensional Poincare duality spaces up to oriented homotopy type. The most important step is a method for describing the Spivak normal fibration and hence the exotic characteristic class.
16 citations
Posted Content•
TL;DR: A survey of classical and recent advances around the existence of exotic differentiable structures on spheres and their connection to stable homotopy theory can be found in this article, with a discussion of the connections between differentiable structure and stable homology.
Abstract: This work surveys classical and recent advances around the existence of exotic differentiable structures on spheres and its connection to stable homotopy theory.
11 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2017 | 1 |
| 2013 | 1 |
| 2012 | 1 |
| 2010 | 3 |
| 2008 | 3 |
| 2006 | 1 |