Let M _ 1 and M _ 2 be two smooth, oriented manifolds of equal dimension and V a smooth, closed, oriented manifold, embedded as a submanifold into both M _ 1 and M _ 2.
12.
Further, Milnor and Wall used the spectral sequence to prove Thom's conjecture on the structure of the oriented cobordism ring : two oriented manifolds are cobordant if and only if their Pontryagin and Stiefel Whitney numbers agree.
13.
If M is a closed, oriented manifold and if M'is obtained from M by removing an open ball, then the connected sum M \ mathrel { \ # }-M is the double of M '.
14.
Once again, if we assume ? ( d + 1 ) can be expressed as an exterior product and that it can be extended into a d + 1-form in a d + 2 dimensional oriented manifold, we can define
15.
If " M " is an oriented manifold, Aut ( " M " ) would be the orientation-preserving automorphisms of " M " and so the mapping class group of " M " ( as an oriented manifold ) would be index two in the mapping class group of " M " ( as an unoriented manifold ) provided " M " admits an orientation-reversing automorphism.
16.
If " M " is an oriented manifold, Aut ( " M " ) would be the orientation-preserving automorphisms of " M " and so the mapping class group of " M " ( as an oriented manifold ) would be index two in the mapping class group of " M " ( as an unoriented manifold ) provided " M " admits an orientation-reversing automorphism.
17.
Consider the ring generated by pairs ( " X ", " V " ) where " V " is a smooth vector bundle on the compact smooth oriented manifold " X ", with relations that the sum and product of the ring on these generators are given by disjoint union and product of manifolds ( with the obvious operations on the vector bundles ), and any boundary of a manifold with vector bundle is 0.
