A new invariant and parametric connected sum of embeddings

Volume 197 / 2007

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


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.


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

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