- #1
Bacle
- 662
- 1
Hi, everyone:
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.
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.