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Integral cohomology rings of weighted Grassmann orbifolds and rigidity properties

Koushik Brahma

Abstract

In this paper, we introduce `Plücker weight vector' and establish the definition of a weighted Grassmann orbifold $\mbox{Gr}_{\mathbf{b}}(k,n)$, corresponding to a Plücker weight vector `$\mathbf{b}$'. We achieve an explicit classification of weighted Grassmann orbifolds up to certain homeomorphism in terms of the Plücker weight vectors. We study the integral cohomology of $\mbox{Gr}_{\bf b}(k,n)$ and provide some sufficient conditions such that the integral cohomology of $\mbox{Gr}_{\bf b}(k,n)$ has no torsion. We describe the integral equivariant cohomology ring of divisive weighted Grassmann orbifolds and compute all the equivariant structure constants with integer coefficients. Eminently, we compute the integral cohomology rings of divisive weighted Grassmann orbifolds explicitly.

Integral cohomology rings of weighted Grassmann orbifolds and rigidity properties

Abstract

In this paper, we introduce `Plücker weight vector' and establish the definition of a weighted Grassmann orbifold , corresponding to a Plücker weight vector `'. We achieve an explicit classification of weighted Grassmann orbifolds up to certain homeomorphism in terms of the Plücker weight vectors. We study the integral cohomology of and provide some sufficient conditions such that the integral cohomology of has no torsion. We describe the integral equivariant cohomology ring of divisive weighted Grassmann orbifolds and compute all the equivariant structure constants with integer coefficients. Eminently, we compute the integral cohomology rings of divisive weighted Grassmann orbifolds explicitly.
Paper Structure (10 sections, 39 theorems, 149 equations)

This paper contains 10 sections, 39 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Building sequences of weighted Grassmann orbifolds
  3. Plücker weight vectors and weighted Grassmann orbifolds
  4. $q$-CW complex structures of weighted Grassmann orbifolds
  5. Classification of weighted Grassmann orbifolds upto certain homeomorphism
  6. Torsion's in the integral cohomology of weighted Grassmann orbifold
  7. Structure constants for the equivariant cohomology rings of divisive weighted Grassmann orbifolds with integer coefficients
  8. Knutson-Tao's results on $H_{T^n}^{*}(\hbox{Gr}(k,n);\mathbb{Z})$
  9. Structure constants for the equivariant cohomology of $\hbox{Gr}_{\bf b}(k,n)$
  10. Structure constants for the integral cohomology ring of divisive weighted Grassmann orbifolds

Key Result

Theorem A

If there exists a homeomorphism $f$ between two weighted Grassmann orbifolds ${\rm{Gr}}_{\bf b}(k,n)$ and ${\rm{Gr}}_{\bf c}(k,n)$ such that the $f$ maps every coordinate element to some coordinate element then ${\bf b}$ and ${\bf c}$ are same up to a scalar multiplication and a permutation.

Theorems & Definitions (85)

  • Theorem A: Theorem \ref{['thm_cohom_rig']}
  • Theorem B: Theorem \ref{['tor_in_2,4']}
  • Theorem C: Theorem \ref{['thm_wei_st_con']}
  • Theorem D: Theorem \ref{['prop_st_con_int']}
  • Theorem E: Theorem \ref{['thm_st_con_ord_coh']}
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • ...and 75 more