Spaces of polynomials as Grassmanians for immersions and embeddings

TL;DR

This paper introduces a new framework called quasitopy to classify certain embeddings and immersions of manifolds into product spaces, using polynomial divisor spaces as Grassmannian analogs, and studies their stability and homotopy properties.

Contribution

It defines quasitopy classes for constrained embeddings and immersions, linking them to polynomial divisor spaces and computing their structure in terms of homotopy and homology.

Findings

Quasitopy classes are computed via homotopy/homology of polynomial divisor spaces.

Quasitopy classes of embeddings stabilize as polynomial degree increases.

Polynomial divisor spaces serve as Grassmannian analogs for classifying manifold embeddings.

Abstract

Let $Y$ be a smooth compact $n$-manifold. We study smooth embeddings and immersions $\beta: M \to \mathbb R \times Y$ of compact $n$-manifolds $M$ such that $\beta(M)$ avoids some a priory chosen closed poset $\Theta$ of {\sf tangent patterns} to the fibers of the obvious projection $\pi: \mathbb R \times Y \to Y$. Then, for a fixed $Y$, we introduce an equivalence relation between such $\beta$'s; it is a crossover between pseudo-isotopies and bordisms. We call this relation {\sf quasitopy}. In the study of quasitopies, the spaces $\mathcal P_d^{\mathbf c\Theta}$ of real univariate polynomials of degree $d$ with real divisors, whose combinatorial patterns avoid a given closed poset $\Theta$, play the classical role of Grassmanians. We compute the quasitopy classes $\mathcal{QT}_d^{\mathsf{emb}}(Y, \mathbf c\Theta)$ of $\Theta$-constrained embeddings $\beta$ in terms of homotopy/homology theory of spaces $Y$ and $\mathcal P_d^{\mathbf c\Theta}$. We prove also that the quasitopies of emeddings stabilize, as $d \to \infty$.