[Paper Review] The symmetric power and étale realisation functors commute

Arnav Tripathy|arXiv (Cornell University)|Feb 4, 2015

Homotopy and Cohomology in Algebraic Topology6 references3 citations

TL;DR

This paper establishes that the symmetric power and étale realization functors commute up to weak equivalence for proper, normal, noetherian, geometrically connected algebraic spaces over a separably closed field. The key result is a weak equivalence between the étale homotopy type of symmetric powers and the symmetric powers of the étale homotopy type, leading to an effective Dold-Thom theorem for étale homotopy and stabilization of étale homotopy groups under symmetric power stabilization maps.

ABSTRACT

We show that under mild hypotheses on a proper algebraic space $X$, the functors of taking its symmetric powers and its étale realisation commute up to weak equivalence. We conclude an effective version of the Dold-Thom theorem for the étale site and discuss the stabilisation results for the natural morphisms of étale homotopy groups $π_k \mathrm{Sym}^n X o π_k \mathrm{Sym}^{n+1} X$ in the context of the Weil conjectures.

Motivation & Objective

• To establish a comparison between symmetric powers and étale homotopy types in algebraic geometry.
• To prove that the natural map from the symmetric power of the étale homotopy type to the étale homotopy type of the symmetric power is a weak equivalence under mild hypotheses.
• To derive an effective Dold-Thom theorem for étale homotopy, relating étale homology to homotopy groups of infinite symmetric powers.
• To analyze the stabilization of étale homotopy groups under the natural inclusion maps $\operatorname{Sym}^n X \to \operatorname{Sym}^{n+1}X$.
• To provide a homotopical foundation for étale homology via symmetric powers, analogous to the classical Dold-Thom theorem in topology.

Proposed method

• Use of Artin and Mazur's étale homotopy theory to associate a pro-homotopy type to algebraic spaces.
• Application of the universal coefficient theorem to relate integral and torsion étale homology in cohomological formulations.
• Employment of Borel-Moore homology and long exact sequences to analyze connectivity of symmetric powers of real affine spaces.
• Inductive argument on dimension using stratifications of symmetric products over simplices to control homology in low degrees.
• Use of the cofiber sequence $\operatorname{Sym}^{n-1}X \to \operatorname{Sym}^n X \to \text{cone}$ to analyze connectivity and stabilization.
• Reduction to the case of spheres via cellular decomposition and symmetric power constructions on $\mathbb{R}^k$.

Experimental results

Research questions

• RQ1 Does the symmetric power functor commute with the étale realization functor up to weak equivalence for proper algebraic spaces?
• RQ2 Can an effective Dold-Thom theorem be formulated for étale homotopy types using symmetric powers?
• RQ3 To what extent do the natural stabilization maps $\operatorname{Sym}^n X \to \operatorname{Sym}^{n+1}X$ induce isomorphisms on étale homotopy groups?
• RQ4 What is the connectivity of the cofiber of the inclusion $\operatorname{Sym}^{n-1}X \to \operatorname{Sym}^n X$ in the étale setting?
• RQ5 How does the homotopical stabilization of symmetric powers in algebraic geometry mirror the topological Dold-Thom theorem?

Key findings

• The natural map $\operatorname{Sym}^n(X_{\text{ét}}) \to (\operatorname{Sym}^n X)_{\text{ét}}$ is a weak equivalence of pro-homotopy types for proper, normal, noetherian, geometrically connected algebraic spaces over a separably closed field.
• The infinite symmetric power $\operatorname{Sym}^\infty X$ realizes the étale homology groups: $\pi_n^{\text{ét}} \operatorname{Sym}^\infty X \simeq \widetilde{H}_n(X_{\text{ét}}, \mathbb{Z})$.
• The stabilization maps $\alpha_n: \operatorname{Sym}^n X \to \operatorname{Sym}^{n+1} X$ induce isomorphisms on $\pi_k^{\text{ét}}$ for $k < n$ and surjections for $k = n$.
• The cofiber of $\operatorname{Sym}^{n-1}S^k \to \operatorname{Sym}^n S^k$ is $S^{nk-1}/S_n$, which is $(n-1)$-connected, implying stabilization in homotopy groups.
• Borel-Moore homology of $\operatorname{Sym}^n \mathbb{R}^k$ vanishes in degrees less than $n$, proven inductively using stratifications and K"unneth-type decompositions.
• The stabilization result holds for symmetric powers of spheres and extends to general algebraic spaces via cellular decomposition and gluing along symmetric products of cells.