**Remarks**

Notice that in the definition of resp. is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set, provided that n>4: The simple structure set is the set of equivalence classes of homotopy equivalences from closed manifolds of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences (i=0,1) are equivalent if there exists a diffeomorphism (or PL-homeomorphism or homeomorphism) such that is homotopic to .

As long as we are dealing with differential manifolds, there is in general no canonical group structure on . If we deal with topological manifolds, it is possible to endow with a preferred structure of an abelian group (see chapter 18 in the book of Ranicki).

Notice that a manifold M is diffeomorphic (or PL-homeomorphic or homeomorphic) to a closed manifold X if and only if there exists a simple homotopy equivalence whose equivalence class is the base point in . Some care is necessary because it may be possible that a given simple homotopy equivalence is not homotopic to a diffeomorphism (or PL-homeomorphism or homeomorphism) although M and X are diffeomorphic (or PL-homeomorphic or homeomorphic). Therefore, it is also necessary to study the operation of the group of homotopy classes of simple self-equivalences of X on .

The basic tool to compute the simple structure set is the surgery exact sequence.

Read more about this topic: Surgery Structure Set

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### Famous quotes containing the word remarks:

“An illustrious individual *remarks* that Mrs. [Elizabeth Cady] Stanton is the salt, Anna Dickinson the pepper, and Miss [Susan B.] Anthony the vinegar of the Female Suffrage movement. The very elements get the “white male” into a nice pickle.”

—Anonymous, U.S. women’s magazine contributor. The Revolution (August 19, 1869)

“The general feeling was, and for a long time remained, that one had several children in order to keep just a few. As late as the seventeenth century . . . people could not allow themselves to become too attached to something that was regarded as a probable loss. This is the reason for certain *remarks* which shock our present-day sensibility, such as Montaigne’s observation, “I have lost two or three children in their infancy, not without regret, but without great sorrow.””

—Philippe Ariés (20th century)

“So, too, if, to our surprise, we should meet one of these morons whose *remarks* are so conspicuous a part of the folklore of the world of the radio—*remarks* made without using either the tongue or the brain, spouted much like the spoutings of small whales—we should recognize him as below the level of nature but not as below the level of the imagination.”

—Wallace Stevens (1879–1955)