# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## A new invariant and parametric connected sum of embeddings

### Tom 197 / 2007

Fundamenta Mathematicae 197 (2007), 253-269 MSC: Primary 57Q35, 57Q37; Secondary 55S15, 55Q91, 57R40. DOI: 10.4064/fm197-0-12

#### Streszczenie

We define an isotopy invariant of embeddings $N\to {{\mathbb R}}^m$ of manifolds into Euclidean space. This invariant together with the $\alpha$-invariant of Haefliger–Wu is complete in the dimension range where the $\alpha$-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 $\alpha$-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 ${(4n+5)/3}\le m\le {(3n+2)/2}$, each preimage of the $\alpha$-invariant injects into a quotient of $H_{3n-2m+3}(N)$, where the coefficients are ${{\mathbb Z}}$ for $m-n$ odd and ${{\mathbb Z}}_2$ for $m-n$ even.

#### Autorzy

• A. SkopenkovDepartment of Differential Geometry
Faculty of Mechanics and Mathematics
Moscow State University
Moscow 119992, Russia
and
Independent University of Moscow
B. Vlasyevskiy, 11, Moscow 119002, Russia
e-mail

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