Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

Total minimality of the unitary groups

Abstract: Dierolf and Schwanengel ([-7]) showed that if X is an infinite diescrete space, then the group F(X) of all bijections f : X ~ X provided with the topology of pointwise convergence is a totally minimal topological group. That was the first example of a (totally) minimal group which is not precompact. Other examples of such groups can be found in [8] (see also [21, w 2]). Infinite groups which does not admit non-discrete Hausdorff group topologies were constructed first by Shelah ([22], assuming CH), and then by Hesse ([14]), and A. Ol'ghanski~ (who noted that a quotient of the Adian's group A(m,n) has the property in question, cf. [1, w Note that all the examples mentioned above are examples of non-Abelian groups. Prodanov ([19]) established the totally minimal Abelian groups are precompact, and recently Prodanov and the author ([20]) proved that all minimal Abelian groups are precompact. The main purpose of the present paper is to show that the unitary group of every real or complex Hilbert space provided with the strong operator topology is a totally minimal topological group. This result gives an affirmative answer to a question posed by I. Prodanov. In Sects. 2 and 3 we study the equivariant (with respect to the action of the unitary group) compactifications of the unit sphere S of an infinite-dimensional Hilbert space. It is shown in Sect. 2 that the unit ball endowed with the weak topology is the greatest equivariant compactification of S. This fact is used in Sect. 3 to describe all equivariant compactification of S. The main theorem is proved in Sect. 4. The proof uses the scheme of the proof of [7, (1)], a generalization of which is discussed in Proposition 4.6.

Cdh descent in equivariant homotopy K-theory

Abstract: We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy K-theory of G-schemes (which we construct as an E-infinity-ring) is stable under arbitrary base change, and we deduce that homotopy K-theory of G-schemes satisfies cdh descent.

Equivariant algebraic cobordism

Abstract: We construct an equivariant algebraic cobordism theory for schemes with an action by a linear algebraic group over a field of characteristic zero.

Quantum cohomology of the Hilbert scheme of points on An-resolutions

Abstract: We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type An singularities. The operators encoding these invariants are expressed in terms of the action of the the affine Lie algebra b gl(n+ 1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/DonaldsonThomas theories of An × P 1 . We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of type D and E.