Application To Classification of Manifolds
The origin and main application of surgery theory lies in the classification of manifolds of dimension greater than four. Loosely, the organizing questions of surgery theory are:
- Is X a manifold?
- Is f a diffeomorphism?
More formally, one must ask whether up to homotopy:
- Does a space X have the homotopy type of a smooth manifold of the same dimension?
- Is a homotopy equivalence f: M → N between two smooth manifolds homotopic to a diffeomorphism?
It turns out that the second ("uniqueness") question is a relative version of a question of the first ("existence") type; thus both questions can be treated with the same methods.
Note that surgery theory does not give a complete set of invariants to these questions. Instead, it is obstruction-theoretic: there is a primary obstruction, and a secondary obstruction called the surgery obstruction which is only defined if the primary obstruction vanishes, and which depends on the choice made in verifying that the primary obstruction vanishes.
Read more about this topic: Surgery Theory
Famous quotes containing the words application to and/or application:
“Preaching is the expression of the moral sentiment in application to the duties of life.”
—Ralph Waldo Emerson (1803–1882)
“I think that a young state, like a young virgin, should modestly stay at home, and wait the application of suitors for an alliance with her; and not run about offering her amity to all the world; and hazarding their refusal.... Our virgin is a jolly one; and tho at present not very rich, will in time be a great fortune, and where she has a favorable predisposition, it seems to me well worth cultivating.”
—Benjamin Franklin (1706–1790)