We define an isotopy invariant of embeddings 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 , where the coefficients are ℤ for m-n odd and ℤ₂ for m-n even.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-12,
author = {A. Skopenkov},
title = {A new invariant and parametric connected sum of embeddings},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {253-269},
zbl = {1145.57019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-12}
}
A. Skopenkov. A new invariant and parametric connected sum of embeddings. Fundamenta Mathematicae, Tome 193 (2007) pp. 253-269. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm197-0-12/