Classification of Manifolds - Dimension 5 and More: Surgery

Dimension 5 and More: Surgery

For more details on this topic, see surgery theory.

In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by surgery theory.

The reason for dimension 5 is that the Whitney trick works in the middle dimension in dimension 5 and more: two Whitney disks generically don't intersect in dimension 5 and above, by general position . In dimension 4, one can resolve intersections of two Whitney disks via Casson handles, which works topologically but not differentiably; see Geometric topology: Dimension for details on dimension.

More subtly, dimension 5 is the cut-off because the middle dimension has codimension more than 2: when the codimension is 2, one encounters knot theory, but when the codimension is more than 2, embedding theory is tractable, via the calculus of functors. This is discussed further below.

Read more about this topic:  Classification Of Manifolds

Famous quotes containing the words dimension and/or surgery:

    Le Corbusier was the sort of relentlessly rational intellectual that only France loves wholeheartedly, the logician who flies higher and higher in ever-decreasing circles until, with one last, utterly inevitable induction, he disappears up his own fundamental aperture and emerges in the fourth dimension as a needle-thin umber bird.
    Tom Wolfe (b. 1931)

    Ever since surgery began, man’s destiny has been to suffer, in order that he might be cured. And no one can change that, gentlemen.
    Jean Scott Rogers, and Robert Day. Mr. Blount (Frank Pettingell)