[Paper Review] Homological stability for complements of closures

Alexander Kupers, Jeremy Miller|arXiv (Cornell University)|Dec 22, 2013

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper proves and generalizes Conjecture F from Vakil and Wood (2012), establishing rational homological stability for complements of discriminant strata in symmetric powers of connected manifolds of dimension at least 2. Using transfer maps and spectral sequences on semisimplicial chain complexes, the authors derive explicit homological stability ranges depending on the manifold's orientability and homological properties.

ABSTRACT

We prove Conjecture F from [VW12] which states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. Moreover, we generalize this conjecture to the case of connected manifolds of dimension at least 2 and give an explicit homological stability range.

Motivation & Objective

• To prove Conjecture F from Vakil and Wood (2012), which posits rational homological stability for complements of closures of strata in symmetric powers of smooth complex varieties.
• To generalize this conjecture from complex varieties to arbitrary connected manifolds of dimension at least 2.
• To provide an explicit homological stability range for the rational homology of these complement spaces.
• To offer a new proof of homological stability for bounded symmetric powers, improving the known stability range.
• To establish a connection between homological stability and stability in the Grothendieck ring of varieties, motivated by motivic zeta functions.

Proposed method

• Define strata $S_{ ho}(M)$, their closures $D_{ ho}(M)$, and complements $W_{ ho}(M)$ in symmetric powers $\mathrm{Sym}_k(M)$, where $\rho$ is a partition of $k$.
• Construct a stabilization map $t: W_{1^j\lambda}(M) \to W_{1^{j+1}\lambda}(M)$ by 'bringing a particle in from infinity' for manifolds with boundary.
• Use transfer maps $\tau: H_i(W_{1^{j+1}\lambda}(M);\mathbb{Q}) \to H_i(W_{1^j\lambda}(M);\mathbb{Q})$ as the primary tool for inducing homology isomorphisms.
• Apply rational singular chains to the resolution of $W_{1^j\lambda}(M)$, forming a semisimplicial chain complex $C_*(\tilde{\mathcal{W}}_\bullet(j);\mathbb{Q})$ with augmentation to $C_*(W_{1^j\lambda}(M);\mathbb{Q})$.
• Use spectral sequences converging to the homology of geometric realizations to compare homology groups across $j$ and $j+1$, relying on the contractibility of fibers in the augmentation map.
• Establish that the transfer map induces an isomorphism on $E^1$-pages of the spectral sequence when $i \leq f^{\mathrm{or}}_{M,\lambda}(j)$ or $i \leq f^{\mathrm{nor}}_{M,\lambda}(j)$, leading to the main stability result.

Experimental results

Research questions

• RQ1 Do the complements of discriminant strata in symmetric powers of connected manifolds of dimension $\geq 2$ exhibit rational homological stability?
• RQ2 Can the stability range for such complements be explicitly quantified in terms of the manifold’s topology and the partition $\lambda$?
• RQ3 Does the transfer map induce rational homology isomorphisms in a range that generalizes and improves upon previous results for bounded symmetric powers?
• RQ4 Is there a structural link between homological stability and stability in the Grothendieck ring of varieties, as suggested by motivic zeta functions?
• RQ5 How does orientability affect the homological stability range, and what role does the vanishing of reduced rational homology groups up to degree $a$ play?

Key findings

• The paper proves Conjecture F: for any irreducible smooth complex variety $X$, the rational homology of $W_{1^j\lambda}(X)$ stabilizes as $j \to \infty$ in fixed degrees.
• For connected manifolds $M$ of dimension $d \geq 2$, rational homology groups $H_i(W_{1^j\lambda}(M);\mathbb{Q})$ are isomorphic for $i \leq f^{\mathrm{or}}_{M,\lambda}(j)$ in the orientable case, and $i \leq f^{\mathrm{nor}}_{M,\lambda}(j)$ in the non-orientable case.
• The stability range is explicitly given by $f^{\mathrm{or}}_{M,\lambda}(j) = (a+1)j - b$, where $a$ is the largest integer such that $\tilde{H}_i(M;\mathbb{Q}) = 0$ for $i \leq a$, and $b$ depends on $\lambda$.
• In the orientable case, the slope of the stability range is $a+1$, with $a < \dim M - 1$, and the range improves upon prior results for bounded symmetric powers.
• For orientable $M$ with $\dim M > 2$, the stability range for $\mathrm{Sym}_k^{\leq c}(M)$ is $i \leq k-1$; for $\dim M = 2$, it is $i \leq \min(k-1, c-4+k)$.
• The augmentation map $||\tilde{\mathcal{W}}_\bullet(j)|| \to \tilde{W}_{1^j\lambda}(M)$ is a weak equivalence, with contractible fibers, which is essential for the spectral sequence argument.