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 .
Read more about this topic: Assembly Map
Famous quotes containing the words importance in, importance and/or geometric:
“There are of course people who are more important than others in that they have more importance in the world but this is not essential and it ceases to be. I have no sense of difference in this respect because every human being comprises the combination form.”
—Gertrude Stein (1874–1946)
“We have been told over and over about the importance of bonding to our children. Rarely do we hear about the skill of letting go, or, as one parent said, “that we raise our children to leave us.” Early childhood, as our kids gain skills and eagerly want some distance from us, is a time to build a kind of adult-child balance which permits both of us room.”
—Joan Sheingold Ditzion (20th century)
“In mathematics he was greater
Than Tycho Brahe, or Erra Pater:
For he, by geometric scale,
Could take the size of pots of ale;
Resolve, by sines and tangents straight,
If bread and butter wanted weight;
And wisely tell what hour o’ th’ day
The clock doth strike, by algebra.”
—Samuel Butler (1612–1680)