Universal Super Vector Bundles

Subject Areas : Statistics

Mohammad Javad Afshari ¹ , Saad Varsaie ²

1 - Iran, Zanjan, Institute for Advanced Studies in Basic Sciences (IASBS)
2 - Associated Professor, Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Nov. 23, 1999 to March 19, 2011.

Received: 2021-02-07 Accepted : 2022-03-13 Published : 2022-06-22

Keywords: &nu, -گراسمنین, رده بندی هموتوپی, برگردان, ابرکلاف برداری,

Abstract :

This article first provides a brief overview of the structure of the classical Grassmann manifold(Grassmannian) and how the universal Grassmann manifold is constructed using maps. Also, the underlying topological space and its sheaf structure are introduced to some extent in a theorem. Then, we enter the topic of super-geometry and a new type of supergrassmannian is introduced by applying odd involution in super ringed space and gluing superdomains. In a similar way to the normal case, the next infinite supergrassmannians and the canonical super vector bundle on it are introduced in the supergeometry. Here our tools mainly include multilinear algebra between supermatrices and their induced mappings, the direct limit in the topology of the underlying spaces and the inverse limit in the structural sheaf of the spaces. Finally, we show that the resulting super bundle is a global member of the category of the super vector bundle; Structures that are used in the classification of super vector bundles and are in proportion to the homotopy classification.

References:

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[2] C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez, The Geometry of Supermanifolds, Kluwer Academic Publisher (1991).

 

[3] M. J. Afshari, S. Varsaie, Homotopy Classification of Super Vector Bundles, arxiv:1802.05506.

 

[4] J. Dugundji, Topology, Allyn and Bacon, Boston (1966).

 

[5] Yu. I. Manin, Gauge Field Theory and Complex Geometry, Springer, New York, 1988.