- #1

Bacle

- 662

- 1

A couple of questions, please:

1) Examples of representative surfaces or curves:

Please let me know if this is a correct definition of a surface representing

H_2(M;Z):

Let M be an orientable m-manifold, Z the integers; m>2 . Let S be an orientable

surface embedded in M. Then H_2(S;Z)=Z . We then say that S represents

H_2(M;Z)~Z if the homomorphism h: Z-->H_2(M;Z) sends 1 --as a generator of Z --

to the homology class of S. specifically, if we have h(1)=alpha ; alpha a homology

class, then there is an embedding i:S-->M , with [i(S)] =alpha.

If this is correct. Anyone know of examples of representative curves or surfaces.?

2)An argument for why non-orientable manifolds have top homology zero, and

for why orientable manifolds have top homology class Z.?.

I have no clue on this one. I know homology zero means that all cycles

are boundaries, but I don't see how this is equivalent to not being orientable.

Thanks in Advance.