Homology stability for symplectic groups

The third homology of symplectic groups and algebraic K-theory

We improve the homology stability range for the 3rd integral homology of symplectic groups over commutative local rings with infinite residue field. As an application, we show that for local commutative rings containing an infinite field of characteristic not 2 the symbol map from Milnor-Witt K-theory to higher Grothendieck-Witt groups is an isomorphism in degrees 2 and 3.

Symplectic homology as Hochschild homology

These are (heavily revised) notes from lectures given at the AMS Algebraic Geometry meeting in Seattle, 2005. The main topic is symplectic homology seen from the point of view of Lefschetz fibrations. Most of the content is speculative, but some supporting evidence is included.

辛群积分同调稳定性范围的研究

On the Homology stability range for symplectic groups

本文改进了辛群的整数同调稳定性范围，并揭示了进一步稳定性的障碍与Milnor-Witt K理论的关系。

The homology groups of some two-step nilpotent Lie algebras associated to symplectic vector spaces

Let H be a symplectic vector space, let V be a vector space, and consider the nilpotent Lie algebra L_H(V) = H \otimes V + S^2(V) with bracket [(h_1 \otimes v_1;a_1),(h_2 \otimes v_2;a_2)] = (0,<h_1,h_2> v_1 v_2) . In this paper, we calculate the Lie algebra homology of L_H(V) as a polynomial functor of V. This has applications to the analysis of the Leray-Serre spectral sequence of the fibrations of moduli spaces of pointed curves M_{1,n}->M_{1,1} and M_{g,n}->M_g.

Homology stability for unitary groups

In this paper homology stability for unitary groups over a ring with finite unitary stable rank is established. Homology stability of symplectic groups and orthogonal groups appears as a special case of our results.

Homology stability for Unitary groups II

In this note the homology stability problem for hyperbolic unitary groups over a local ring with an infinite residue field is studied.