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Keywords = spaces of polynomials with constrained real divisors

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48 pages, 944 KB

Open AccessArticle

Spaces of Polynomials as Grassmanians for Immersions and Embeddings

by Gabriel Katz

Int. J. Topol. 2025, 2(3), 9; https://doi.org/10.3390/ijt2030009 - 24 Jun 2025

Viewed by 675

Abstract

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions

β : M \to R \times Y

of compact n-manifolds M such that

β (M)

avoids some priory chosen closed poset

Θ

of tangent patterns to [...] Read more.

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions

β : M \to R \times Y

of compact n-manifolds M such that

β (M)

avoids some priory chosen closed poset

Θ

of tangent patterns to the fibers of the obvious projection

π : R \times Y \to Y

. Then, for a fixed Y, we introduced an equivalence relation between such

β

’s; creating a crossover between pseudo-isotopies and bordisms. We called this relation quasitopy. In the presented study of quasitopies, the spaces

P_{d}^{c Θ}

of real univariate polynomials of degree d with real divisors, whose combinatorial patterns avoid a given closed poset

Θ

, play the classical role of Grassmanians. We computed the quasitopy classes

Q_{d}^{emb} (Y, c Θ)

Θ

-constrained embeddings

β

in terms of homotopy/homology theory of spaces Y and

P_{d}^{c Θ}

. We proved also that the quasitopies of embeddings stabilize, as

d \to \infty

. Full article

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