A new invariant and parametric connected sum of embeddings 

A. Skopenkov 

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

We introduce an isotopy invariant of embeddings of manifolds into 
Euclidean space (which is analogous to the Sato-Levine invariant of knots, the 
Hudson-Habegger obstruction to embedding disks and Fenn-Rolfsen-Koschorke-Kirk 
beta-invariant of link maps). 
This invariant together with the Haefliger-Wu invariant is complete in the 
dimension range where the Haefliger-Wu invariant could be incomplete.
We also introduce parametric connected sum of certain embeddings 
(analogous to surgery). 

This allows to obtain estimation of isotopy classes of embeddings $N\to R^m$ 
stronger than in [1].
For 4-manifold N and m=7 this is done in terms of an exact sequence 
involving the $H_2(N;Z)$-valued Whitney invariant and an explicitly 
constructed action of $H_1(N;Z\oplus Z_2)$ on the set of embeddings.  

Reference: A. Skopenkov. 
A new invariant and parametric connected sum of embeddings
http://arxiv.org/abs/math/0509621