Change in Homology from Change in Coefficients

[Bacle]

Hi, everyone:

If we change the coefficients used to calculate homology, the universal coefficient
theorem tells us how the homology changes. Still, is there a way of knowing whether
a specific (non-trivial) cycle under certain coefficient is still a cycle after the coefficient
change?

Thanks.

 

[Bacle]

I think my post was not very clear: I am curious as to whether torsion is intrinsic
to a space, in that if a space X has torsion, then homology with _any coefficients_
would also have torsion. I was thinking of examples like that of the torus T, in which
torsion appears when we use coefficients with torsion, e.g, Z/2, but not otherwise.

 

[lavinia]

I think my post was not very clear: I am curious as to whether torsion is intrinsic
to a space, in that if a space X has torsion, then homology with _any coefficients_
would also have torsion. I was thinking of examples like that of the torus T, in which
torsion appears when we use coefficients with torsion, e.g, Z/2, but not otherwise.

Torsion is not intrinsic. For instance the Z/2 homology of the torus is all torsion but its Z homology is free abelian.

For unorientable manifolds such as the Klein bottle the top Z homology is zero but the top z/2 homology is Z/2.

If the coefficient ring is the rational numbers there is never any torsion.

 

[Bacle]

Torsion is not intrinsic. For instance the Z/2 homology of the torus is all torsion but its Z homology is free abelian.

For unorientable manifolds such as the Klein bottle the top Z homology is zero but the top z/2 homology is Z/2.

If the coefficient ring is the rational numbers there is never any torsion.

But then does torsion really tell us something useful about the topology of a space, or does it just reveal some algebraic property of the coefficients used?

 

[lavinia]

But then does torsion really tell us something useful about the topology of a space, or does it just reveal some algebraic property of the coefficients used?

yes. Unorientablility is intrinsinc. In this sense you might say that torsion is intrinsic though it is only picked up in certain coefficient rings and not in others.

In the real projective plane all cycles in all dimensions are torsion in that their Z-boundaries are double another chain. So over Z they are not cycles but over Z/2 they are.

If you look at it this way then torsion is an intrinsic topological property. maybe the right way to look at it in homology is to say that there is intrinsic 2-torsion when a chain is not a Z-cycle but is a z/2 cycle. So I guess I should take back my statement that torsion is not intrinsic.

Similarly you could have Z-chains that are not cyles but are cycles in Z/nZ. This would be intrinsic n-torsion.

In cohomology the first Stiefel-Whitney class is a Z/2 cocycle that is not zero if an only if the vector bundle is not orientable. For the tangent bundle this means that it is not zero if and only if the manifold is not orientable.