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A new invariant and parametric connected sum of embeddings

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Skopenkov A.

Fundamenta Mathematicae

2007

tom 197

nr 1

253-269

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.

On the generalized Massey–Rolfsen invariant for link maps

100%

Skopenkov A.

Fundamenta Mathematicae

2000

tom 165

nr 1

1-15

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 $π^{m-1}(K^∼)≅⊕_{i < j} π^S_{p_i + p_j - m + 1}$.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Skopenkov A.

1 2007

1 2000

Wyniki wyszukiwania

A new invariant and parametric connected sum of embeddings

On the generalized Massey–Rolfsen invariant for link maps