# surgery exact sequence in a sentence

- So the
*surgery exact sequence* is an exact sequence in the following sense: - The structure set and the surgery obstruction map are brought together in the
*surgery exact sequence*. - In important cases, the smooth or topological structure set can be computed by means of the
*surgery exact sequence*. - From the
*surgery exact sequence* for S ^ n for n \ geq 5 in the topological category we see that - In the topological category, the
*surgery exact sequence* is the long exact sequence induced by a fibration sequence of spectra. - It's difficult to find
*surgery exact sequence* in a sentence. - In fact, these calculations can be formulated in a modern language in terms of the
*surgery exact sequence* as indicated here. - 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. - 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. - 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. - 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. - 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 ).