Assembly Map - Importance in Geometric Topology

Importance in Geometric Topology

Assembly maps are studied in geometric topology mainly for the two functors, algebraic L-theory of, and, algebraic K-theory of spaces of . In fact, the homotopy fibers of both assembly maps have a direct geometric interpretation when is a compact topological manifold. Therefore knowledge about the geometry of compact topological manifolds may be obtained by studying - and -theory and their respective assembly maps.

In the case of -theory, the homotopy fiber of the corresponding assembly map, evaluated at a compact topological manifold, is homotopy equivalent to the space of block structures of . Moreover, the fibration sequence

induces a long exact sequence of homotopy groups which may be identified with the surgery exact sequence of . This may be called the fundamental theorem of surgery theory and was developed subsequently by Browder, Novikov, Sullivan, Wall, Quinn, and Ranicki.

For -theory, the homotopy fiber of the corresponding assembly map is homotopy equivalent to the space of stable h-cobordisms on . This fact is called the stable parametrized h-cobordism theorem, proven by Waldhausen-Jahren-Rognes. It may be viewed as a parametrized version of the classical theorem which states that equivalence classes of h-cobordisms on are in 1-to-1 correspondence with elements in the Whitehead group of .