Talk:Classification of manifolds

Question
In what sense are higher dimensional manifold "classified by surgery theory"? This is mentioned several times and is potentially misleading. Jjauregui (talk) 03:18, 3 October 2008 (UTC)

Bordism groups
The following line is somewhat misleading:

"Cobordism groups (the bordism groups of a point) are computed, but the bordism groups of a space (such as MO * (M)) are generally not."

There are several types of bordism. If we are talking about unoriented, oriented, or complex bordism, usually denoted MO, MSO and MU respectively, then the first statement (about the bordism groups of a point being known) is true. However if we are talking about framed bordism or symplectic bordism, the bordism groups of a point are not known, eg in the framed case they are the stable homotopy groups of spheres.

The second statement also needs modifying - the unoriented bordism groups MO_*(M) of a space M are known once you know the mod 2 cohomology groups of the space (which are computable if M is homotopy equivalent to CW complex, eg a manifold). —Preceding unsigned comment added by 129.215.104.21 (talk) 13:23, 9 October 2009 (UTC)