Classification of manifolds and splitting
ICTP (Main Building Seminar Room)
Professor Yuri Muranov
(Vitebsk State University, Belarus)
Let $Y\subset X$ be a closed manifold pair. A simple homotopy equivalence $f:M\to X$ of n-manifolds splits along the submanifold $Y$ if it is homotopy equivalent to a map which is a simple homotopy equivalence on the transversal preimages of the submanifold and its compliment. In this situation the splitting obstruction groups $LS_*$ are well defined and provide an evident obstruction to existence a homeomorphism in the homotopy class of a simple homotopy equivalence $f$. The splitting obstruction groups closely relates to detecting what elements of the Wall surgery obstruction group belong to the image of the assembly map. From the geometrical point of view the splitting obstruction is an obstruction to do surgery inside the ambient manifold. We describe relations between surgery on manifold and the splitting problem and give a generalization of the splitting theory to the case of filtered manifolds.
We discuss the applications of the splitting theory to the classification of closed manifolds.
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