surgery exact sequence in a sentence

1. So the surgery exact sequence is an exact sequence in the following sense:
2. The structure set and the surgery obstruction map are brought together in the surgery exact sequence.
3. In important cases, the smooth or topological structure set can be computed by means of the surgery exact sequence.
4. From the surgery exact sequence for S ^ n for n \ geq 5 in the topological category we see that
5. In the topological category, the surgery exact sequence is the long exact sequence induced by a fibration sequence of spectra.
6. It's difficult to find surgery exact sequence in a sentence.
7. In fact, these calculations can be formulated in a modern language in terms of the surgery exact sequence as indicated here.
8. The idea of the surgery exact sequence is implicitly present already in the original article of Kervaire and Milnor on the groups of homotopy spheres.
9. Moreover, the surgery exact sequence can be identified with the algebraic surgery exact sequence of Ranicki which is an exact sequence of abelian groups by definition.
10. Moreover, the surgery exact sequence can be identified with the algebraic surgery exact sequence of Ranicki which is an exact sequence of abelian groups by definition.
11. In practice one has to proceed case by case, for each manifold \ mathcal { } X it is a unique task to determine the surgery exact sequence, see some examples below.
12. Also note that there are versions of the surgery exact sequence depending on the category of manifolds we work with : smooth ( DIFF ), PL, or topological manifolds and whether we take Whitehead torsion into account or not ( decorations s or h ).