A new invariant and parametric connected sum of embeddings

A. Skopenkov

Fundamenta Mathematicae (2007)

  • Volume: 197, Issue: 1, page 253-269
  • ISSN: 0016-2736

Abstract

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We define an isotopy invariant of embeddings N → ℝ m of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows us to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n-2m+2)-connected n-manifold N with (4n+5)/3 ≤ m ≤ (3n+2)/2, each preimage of the α-invariant injects into a quotient of H 3 n - 2 m + 3 ( N ) , where the coefficients are ℤ for m-n odd and ℤ₂ for m-n even.

How to cite

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A. Skopenkov. "A new invariant and parametric connected sum of embeddings." Fundamenta Mathematicae 197.1 (2007): 253-269. <http://eudml.org/doc/283147>.

@article{A2007,
abstract = {We define an isotopy invariant of embeddings $N → ℝ^\{m\}$ of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows us to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n-2m+2)-connected n-manifold N with (4n+5)/3 ≤ m ≤ (3n+2)/2, each preimage of the α-invariant injects into a quotient of $H_\{3n-2m+3\}(N)$, where the coefficients are ℤ for m-n odd and ℤ₂ for m-n even.},
author = {A. Skopenkov},
journal = {Fundamenta Mathematicae},
keywords = {embedding; deleted product; self-intersection; isotopy; Haefliger-Wu invariant},
language = {eng},
number = {1},
pages = {253-269},
title = {A new invariant and parametric connected sum of embeddings},
url = {http://eudml.org/doc/283147},
volume = {197},
year = {2007},
}

TY - JOUR
AU - A. Skopenkov
TI - A new invariant and parametric connected sum of embeddings
JO - Fundamenta Mathematicae
PY - 2007
VL - 197
IS - 1
SP - 253
EP - 269
AB - We define an isotopy invariant of embeddings $N → ℝ^{m}$ of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows us to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n-2m+2)-connected n-manifold N with (4n+5)/3 ≤ m ≤ (3n+2)/2, each preimage of the α-invariant injects into a quotient of $H_{3n-2m+3}(N)$, where the coefficients are ℤ for m-n odd and ℤ₂ for m-n even.
LA - eng
KW - embedding; deleted product; self-intersection; isotopy; Haefliger-Wu invariant
UR - http://eudml.org/doc/283147
ER -

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