Splitting a simple homotopy equivalence along a submanifold with filtration

A simple homotopy equivalence $f\colon M^n\to X^n$ of manifolds splits along a submanifold $Y\subset X$ if it is homotopic to a map that is a simple homotopy equivalence on the transversal preimage of the submanifold and on the complement of this preimage. The problem of splitting along a submanifold with filtration is a natural generalization of this problem. In this paper we define groups $\mathit{LSF}_*$ of obstructions to splitting along a submanifold with filtration and describe their properties. We apply the results obtained to the problem of the realization of surgery and splitting obstructions by maps of closed manifolds and consider several examples.
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