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Cohomology rings of oriented Grassmann manifolds $\widetilde G_{2^t,4}$

Uroš A. Colović, Milica Jovanović, Branislav I. Prvulović

Abstract

We give a description of the mod 2 cohomology algebra of the oriented Grassmann manifold $\widetilde G_{2^t,4}$ as the quotient of a polynomial algebra by a certain ideal. In the process we find a Gröbner basis for that ideal, which we then use to exhibit an additive basis for $H^*(\widetilde G_{2^t,4};\mathbb Z_2)$.

Cohomology rings of oriented Grassmann manifolds $\widetilde G_{2^t,4}$

Abstract

We give a description of the mod 2 cohomology algebra of the oriented Grassmann manifold as the quotient of a polynomial algebra by a certain ideal. In the process we find a Gröbner basis for that ideal, which we then use to exhibit an additive basis for .

Paper Structure

This paper contains 12 sections, 16 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction and statement of main results
  2. Preliminaries
  3. Cohomology of Grassmann manifolds
  4. Gröbner bases
  5. A generating set for the algebra $H^*(\widetilde{G}_{2^t,4})$
  6. Generators for $H^*(W_{3,1}^{2^t})$
  7. Generators for $H^*(\widetilde{G}_{2^t,4})$
  8. Cohomology algebra $H^*(\widetilde{G}_{2^t,4})$
  9. Gröbner basis for $I$
  10. Proof of Theorem \ref{['Cohomology_G_2^t,4']}
  11. Some additional facts about $H^*(\widetilde{G}_{2^t,4})$
  12. The case $t=3$

Key Result

Theorem 1.1

Let $t\geqslant3$. There is an isomorphism of graded algebras where $|w_i| = i$, $i=2,3,4$, $|a_{2^t-4}|=2^t-4$, and $P$ and $Q$ are some polynomials in $w_2$, $w_3$ and $w_4$.

Figures (1)

  • Figure 1: $E_\infty$-page of the spectral sequence for $S^3\rightarrow W_{3,1}^{2^t} \xrightarrow{sp} \widetilde{G}_{2^t,4}$

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 19 more