[Paper Review] Cellular $\\mathbb A^1$-homology and the motivic version of Matsumoto's theorem

Fabien Morel, Anand Sawant|arXiv (Cornell University)|Jul 29, 2020

Advanced Algebra and Geometry4 citations

TL;DR

This paper introduces cellular $Ä^1$-homology, a new computable homology theory for smooth schemes with cellular structures, to compute the $Ä^1$-fundamental group of split reductive groups over arbitrary fields. Using this theory, it establishes the motivic version of Matsumoto's theorem, showing that the $Ä^1$-fundamental sheaf of a split, semisimple, simply connected algebraic group is isomorphic to Milnor $K$-theory ${\mathbf{K}}^\mathrm{M}_2$ or Milnor-Witt $K$-theory ${\mathbf{K}}^\mathrm{MW}_2$ depending on the type, unifying and generalizing classical results in algebraic topology and algebraic $K$-theory.

ABSTRACT

We define a new version of $\\mathbb A^1$-homology, called cellular $\\mathbb A^1$-homology, for smooth schemes over a field that admit an increasing filtration by open subschemes with cohomologically trivial closed strata. We provide several explicit computations of cellular $\\mathbb A^1$-homology and use them to determine the $\\mathbb A^1$-fundamental group of a split reductive group over an arbitrary field, thereby obtaining the motivic version of Matsumoto's theorem on universal central extensions of split, semisimple, simply connected algebraic groups. As applications, we uniformly explain and generalize results due to Brylinski-Deligne and Esnault-Kahn-Levine-Viehweg, determine the isomorphism classes of central extensions of such an algebraic group by an arbitrary strictly $\\mathbb A^1$-invariant sheaf and also reprove classical results of E. Cartan on homotopy groups of complex Lie groups.

Motivation & Objective

• To define a new, computable version of $Ä^1$-homology—cellular $Ä^1$-homology—for smooth schemes with cellular filtrations.
• To compute the $Ä^1$-fundamental sheaf of groups ${\bm{\pi}}^{\u00c4^1}_1(G)$ for split reductive groups over an arbitrary field.
• To establish the motivic version of Matsumoto's theorem on universal central extensions of split, semisimple, simply connected algebraic groups.
• To unify and generalize results of Brylinski-Deligne and Esnault-Kahn-Levine-Viehweg on central extensions.
• To reprove classical results of E. Cartan on homotopy groups of complex Lie groups using motivic methods.

Proposed method

• Defining cellular $Ä^1$-homology as the homology of a cellular $Ä^1$-chain complex associated to a scheme with a cellular structure.
• Using the $Ä^1$-homotopy purity theorem and the structure of $Ä^1$-homotopy sheaves over perfect fields to analyze the fundamental group.
• Constructing the cellular $Ä^1$-chain complex from the Bruhat decomposition of a split, semisimple, simply connected algebraic group.
• Computing the differential in low degrees of the cellular $Ä^1$-chain complex using explicit models of punctured affine spaces and their suspensions.
• Establishing an isomorphism between the $Ä^1$-fundamental sheaf and Milnor $K$-theory or Milnor-Witt $K$-theory via comparison with singular homology of complex points.
• Using spectral sequences and duality in motivic cohomology to verify the vanishing and rank of cohomology groups in low degrees.

Experimental results

Research questions

• RQ1 What is the $Ä^1$-fundamental sheaf of groups ${\bm{\pi}}^{\u00c4^1}_1(G)$ for a split reductive group $G$ over an arbitrary field $k$?
• RQ2 How can cellular $Ä^1$-homology be used to compute this fundamental sheaf in a way that generalizes classical results?
• RQ3 What is the motivic analogue of Matsumoto's theorem on universal central extensions in the context of algebraic groups?
• RQ4 How does the cellular $Ä^1$-homology of the flag variety $G/B$ relate to the motivic cohomology of $G$?
• RQ5 Can the classical result of E. Cartan on $Ï _3(G({\mathbb{C}}))$ be rederived using motivic homotopy theory?

Key findings

• The $Ä^1$-fundamental sheaf of groups ${\bm{\pi}}^{\u00c4^1}_1(G)$ for a split reductive group $G$ fits into a short exact sequence $1 \to {\bm{\pi}}^{\u00c4^1}_1(G_{\rm sc}) \to {\bm{\pi}}^{\u00c4^1}_1(G) \to \mu \to 1$, where $\mu$ is the kernel of the universal cover.
• For a split, semisimple, almost simple, simply connected algebraic group $G$, the $Ä^1$-fundamental sheaf is isomorphic to ${\mathbf{K}}^\mathrm{M}_2$ if $G$ is not of symplectic type, and to ${\mathbf{K}}^\mathrm{MW}_2$ if it is of symplectic type.
• The cellular $Ä^1$-homology of the flag variety $G/B$ in low degrees is computed explicitly, with the degree 2 differential image determined by root system data.
• The complex points of the cellular $Ä^1$-chain complex are shown to compute the singular homology of $G({\mathbb{C}})$, and the spectral sequence for $H^*(G;{\mathbf{K}}^\mathrm{M}_2)$ agrees with the Serre spectral sequence after reindexing.
• The morphism $H^1(G;{\mathbf{K}}^\mathrm{M}_2) \to H^1(G({\mathbb{C}});{\mathbb{Z}})$ is an isomorphism, and $H^i(G;{\mathbf{K}}^\mathrm{M}_2) = 0$ for $i \geq 2$, confirming the motivic variant of Cartan's theorem.
• The dual of the cellular $Ä^1$-chain complex in degree 3 is shown to have homology isomorphic to ${\mathbb{Z}}$, proving $\pi_3(G({\mathbb{C}})) \cong {\mathbb{Z}}$ via motivic methods.