In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30

Transformation of Groups

Equivariant cobordism for torus actions

We obtain an explicit presentation of the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation of the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational cobordism rings of wonderful symmetric varieties of minimal rank.

Equivariant Cobordism of Flag Varieties and of Symmetric Varieties

We associate a bivariant theory to any suitable oriented Borel–Moore homology theory on the category of algebraic schemes or the category of algebraic G-schemes. Applying this to the theory of algebraic cobordism yields operational cobordism rings and operational G-equivariant cobordism rings associated to all schemes in these categories. In the case of toric varieties, the operational T-equivariant cobordism ring may be described as the ring of piecewise graded power series on the fan with coefficients in the Lazard ring.

/pdf/bivariant-algebraic-cobordism-2ynn39qava.pdf

Bivariant algebraic cobordism

We obtain an explicit presentation for the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation for the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational equivariant cobordism rings of wonderful symmetric varieties of minimal rank.

https://www.hse.ru/data/2014/03/20/1306969253/Kiritchenko_Krishna.pdf

Equivariant cobordism of flag varieties and of symmetric varieties

The Algebraic Cobordism Ring of Toric Varieties

We describe the equivariant cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties.

Cobordism ring of toric varieties

Let $X(Q,\Lambda)$ be a quasitoric manifold associated to a simple convex polytope $Q$ and characteristic function $\Lambda$. Let $T\cong (\mathbb{S}^1)^n$ denote the compact $n$-torus acting on $X=X(Q,\Lambda)$. The main aim of this article is to give a presentation of the $T$-equivariant $K$-ring of $X$, as a Stanley-Reisner ring over $K^*(pt)$. We also derive the presentation for the ordinary $K$-ring of $X$.

Equivariant $K$-theory of quasitoric manifolds

Let $X$ be a torus manifold with locally standard action of a compact torus $T$ of half the dimension and orbit space a homology polytope. Smooth complete complex toric varieties and quasi-toric manifolds are examples of torus manifolds. Consider a principal bundle with total space $E$ and base $B$ with fibre and structure group $T$. Let $E(X)$ denote the total space of the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as an algebra over the singular cohomology ring of $B$ and a presentation of the topological $K$-ring of $E(X)$ as an algebra over the topological $K$-ring of $B$. These are relative versions of the results of M. Masuda and T. Panov [13] on the cohomology ring of a torus manifold and P. Sankaran [14] on the topological $K$-ring of a torus manifold. Further, they extend the results due to P. Sankaran and V. Uma [15] on the cohomology ring and topological $K$-ring of toric bundles with fibre a smooth projective toric variety, to a toric bundle with fibre any smooth complete toric variety.

Cohomology of torus manifold bundles

Toric orbifolds are topological generalization of projrctive toric varieties associated to simplicial fans. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure an invariant CW-structure on it. In this paper we study equivariant K-theory, cobordism and cohomology theories of divisive toric orbifolds. These extend the similar results on divisive weighted projective spaces.

V. Uma

Papers

The Algebraic Cobordism Ring of Toric Varieties

Cobordism ring of toric varieties

Equivariant $K$-theory of quasitoric manifolds

Cohomology of torus manifold bundles

Equivariant $K$-theory and cobordism rings of divisive toric varieties and toric orbifolds