On the generalized Massey–Rolfsen invariant for link maps

A. Skopenkov

ArticleOriginal scientific text

Title

On the generalized Massey–Rolfsen invariant for link maps

Authors ¹

Affiliations

Department of Differential Geometry, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia 119899

Abstract

For

K = K_{1} ⊔ ... ⊔ K_{s}

and a link map

f : K \to ℝ^{m}

let

K^{\sim} = ⊔_{i < j} K_{i} \times K_{j}

, define a map

f^{\sim} : K^{\sim} \to S^{m - 1}

by

f^{\sim} (x, y) = \frac{f x - f y}{| f x - f y |}

and a (generalized) Massey-Rolfsen invariant

α (f) \in π^{m - 1} (K)

to be the homotopy class of

f^{\sim}

. 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 \to ℝ^{m}

up to link concordance to

π^{m - 1} (K^{\sim})

. 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^{\sim}) ≅ \oplus_{i < j} π^{S}_{p_{i} + p_{j} - m + 1}

.

Keywords

ENG

deleted product, Massey-Rolfsen invariant, link maps, link homotopy, stable homotopy group, double suspension, codimension two, highly connected manifolds

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Title

On the generalized Massey–Rolfsen invariant for link maps

Affiliations

Abstract

Keywords

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