- #1
Bacle
- 662
- 1
Hi, Everybody:
I am trying to understand torsion and relative cycles in a more geometric way; I
think I understand some of the machinery behind relative cycles (i.e., the LES, and
the induced maps.), and I understand that by ,e .g., Poincare duality, in order to have
torsion in homology, we must have algebraic torsion ( I don't think this came out too
clearly, tho I hope clearly-enough. Please let me know o.wise).
So let me start with this case that may cover both issues:
Say we have the Mobius band M, with boundary component B, and we consider
H_2(M,B;Z)=Z/2 . I can figure this from the LES, but I am having trouble forming a mental
picture of what the actual torsion cycles would be like. I imagine the connection between
algebra and geometry is the obvious one :that going one around will give us a
nonbounding cycle, but going twice around will give us a boundary.
2) How about an example of relative cycles,in (X,A) both trivial ones and bounding ones,
please.?. I understand a cycle in X is a relative boundary if its boundary is fully
contained in A. Still, I am having trouble considering actual geometric examples.
Thanks in Advance.
I am trying to understand torsion and relative cycles in a more geometric way; I
think I understand some of the machinery behind relative cycles (i.e., the LES, and
the induced maps.), and I understand that by ,e .g., Poincare duality, in order to have
torsion in homology, we must have algebraic torsion ( I don't think this came out too
clearly, tho I hope clearly-enough. Please let me know o.wise).
So let me start with this case that may cover both issues:
Say we have the Mobius band M, with boundary component B, and we consider
H_2(M,B;Z)=Z/2 . I can figure this from the LES, but I am having trouble forming a mental
picture of what the actual torsion cycles would be like. I imagine the connection between
algebra and geometry is the obvious one :that going one around will give us a
nonbounding cycle, but going twice around will give us a boundary.
2) How about an example of relative cycles,in (X,A) both trivial ones and bounding ones,
please.?. I understand a cycle in X is a relative boundary if its boundary is fully
contained in A. Still, I am having trouble considering actual geometric examples.
Thanks in Advance.