Spaces of Polynomials as Grassmanians for Immersions and Embeddings
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ijt-02-00009.pdf
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Author(s)
Katz, Gabriel
Date Issued
June 24, 2025
Journal
International Journal of Topology
Publisher
Multidisciplinary Digital Publishing Institute
Citation
Katz, G. Spaces of Polynomials as Grassmanians for Immersions and Embeddings. Int. J. Topol. 2025, 2, 9.
Version
Final published version
Abstract
: Let Y be a smooth compact n-manifold. We studied smooth embeddings and
immersions β : M → R × 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 × Y → 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
cΘ
d
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 Qemb
d
(Y, cΘ) of Θ-constrained embeddings
β in terms of homotopy/homology theory of spaces Y and P
cΘ
d
. We proved also that the
quasitopies of embeddings stabilize, as d → ∞.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.3390/ijt2030009
