A new invariant and parametric connected sum of embeddings
Abstract
We define an isotopy invariant of embeddings N -> Rm 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 to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. For the piecewise-linear category, a (3n-2m+2)-connected n-manifold N and (4n+4)/3 < m < (3n+3)/2 each preimage of α-invariant injects into a quotient of H3n-2m+3(N), where the coefficients are Z for m-n odd and Z2 for m-n even.
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