On the generalized Massey–Rolfsen invariant for link maps

A. Skopenkov

Fundamenta Mathematicae (2000)

  • Volume: 165, Issue: 1, page 1-15
  • ISSN: 0016-2736

Abstract

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For K = K 1 ⊔ . . . ⊔ K s and a link map f : K → ℝ m let K ∼ = ⊔ i < j K i × K j , define a map f ∼ : K ∼ → S m - 1 by f ∼ ( x , y ) = ( f x - f y ) / | f x - f y | and a (generalized) Massey-Rolfsen invariant α ( f ) ∈ π m - 1 ( K ) to be the homotopy class of f ∼ . We prove that for a polyhedron K of dimension ≤ m - 2 under certain (weakened metastable) dimension restrictions, α is an onto or a 1 - 1 map from the set of link maps f : K → ℝ m up to link concordance to π m - 1 ( K ∼ ) . If K 1 , . . . , K s are closed highly homologically connected manifolds of dimension p 1 , . . . , p s (in particular, homology spheres), then π m - 1 ( K ∼ ) ≅ ⊕ i < j π p i + p j - m + 1 S .

How to cite

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Skopenkov, A.. "On the generalized Massey–Rolfsen invariant for link maps." Fundamenta Mathematicae 165.1 (2000): 1-15. <http://eudml.org/doc/212458>.

@article{Skopenkov2000,
abstract = {For $K = K_1⊔...⊔K_s$ and a link map $f:K → ℝ^m$ let $K^∼ = ⊔_\{i < j\} K_i × K_j$, define a map $f^∼ : K^∼ → S^\{m - 1\}$ by $f^∼(x, y) = (fx - fy)/|fx - fy|$ and a (generalized) Massey-Rolfsen invariant $α(f) ∈ π^\{m - 1\}(K)$ to be the homotopy class of $f^∼$. We prove that for a polyhedron K of dimension ≤ m - 2 under certain (weakened metastable) dimension restrictions, α is an onto or a 1 - 1 map from the set of link maps $f:K → ℝ^m$ up to link concordance to $π^\{m - 1\}(K^∼)$. If $K_1,...,K_s$ are closed highly homologically connected manifolds of dimension $p_1,...,p_s$ (in particular, homology spheres), then $π^\{m-1\}(K^∼)≅⊕_\{i < j\} π^S_\{p_i + p_j - m + 1\}$.},
author = {Skopenkov, A.},
journal = {Fundamenta Mathematicae},
keywords = {deleted product; Massey-Rolfsen invariant; link maps; link homotopy; stable homotopy group; double suspension; codimension two; highly connected manifolds; link map; link concordance; cohomotopy},
language = {eng},
number = {1},
pages = {1-15},
title = {On the generalized Massey–Rolfsen invariant for link maps},
url = {http://eudml.org/doc/212458},
volume = {165},
year = {2000},
}

TY - JOUR
AU - Skopenkov, A.
TI - On the generalized Massey–Rolfsen invariant for link maps
JO - Fundamenta Mathematicae
PY - 2000
VL - 165
IS - 1
SP - 1
EP - 15
AB - For $K = K_1⊔...⊔K_s$ and a link map $f:K → ℝ^m$ let $K^∼ = ⊔_{i < j} K_i × K_j$, define a map $f^∼ : K^∼ → S^{m - 1}$ by $f^∼(x, y) = (fx - fy)/|fx - fy|$ and a (generalized) Massey-Rolfsen invariant $α(f) ∈ π^{m - 1}(K)$ to be the homotopy class of $f^∼$. We prove that for a polyhedron K of dimension ≤ m - 2 under certain (weakened metastable) dimension restrictions, α is an onto or a 1 - 1 map from the set of link maps $f:K → ℝ^m$ up to link concordance to $π^{m - 1}(K^∼)$. If $K_1,...,K_s$ are closed highly homologically connected manifolds of dimension $p_1,...,p_s$ (in particular, homology spheres), then $π^{m-1}(K^∼)≅⊕_{i < j} π^S_{p_i + p_j - m + 1}$.
LA - eng
KW - deleted product; Massey-Rolfsen invariant; link maps; link homotopy; stable homotopy group; double suspension; codimension two; highly connected manifolds; link map; link concordance; cohomotopy
UR - http://eudml.org/doc/212458
ER -

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