Def. of (mod2) reduction of w in H_2(M,Z)

[Bacle]

Hi, All:

Just curious as to the definition of the "mod2 reduction of a homology class".

The context is that an element w in H₂(M⁴,Z) is called

characteristic if "its mod reduction [w]₂ is Poincare-dual to the

Stiefel-Whitney class w₂ in H²(M⁴,Z),

where M⁴ is a 4-manifold. Does the reduction just mean that we

start with a Z-chain , and then each coefficient term in the chain is evaluated

mod2?

Thanks.

 

[quasar987]

Yes. If M is an R-module and N a submodule, then it is easy to see that for any topological space X, the morphism of graded R-modules

C_*(X;M)\rightarrow C_*(X;M/N): xc\mapsto [x]c

is a chain morphism. So it passes to homology and the resulting morphism

H_*(X;M)\rightarrow H_*(X;M/N): x[z]\mapsto [x][z]

is called "mod N reduction of the homology".

 

[lavinia]

Hi, All:

Just curious as to the definition of the "mod2 reduction of a homology class".

The context is that an element w in H₂(M⁴,Z) is called

characteristic if "its mod reduction [w]₂ is Poincare-dual to the

Stiefel-Whitney class w₂ in H²(M⁴,Z),

where M⁴ is a 4-manifold. Does the reduction just mean that we

start with a Z-chain , and then each coefficient term in the chain is evaluated

mod2?

Thanks.

A Z chain becomes a Z/2 chain by coefficient projection , Z -> Z/2.

The exact sequence of coefficients 0 -> Z -> Z -> Z/2 -> 0 gives a long exact sequence of homology groups. I think the connecting homomorphism is called the Bockstein homomorphism.

 

[Bacle]

Thanks, both.

I guess then that a characteristic surface would then be Poincare-

dual to the Stiefel-Whitney 2-class, right, i.e., a cocycle w^

with : w^\/ c = c\/c , where \/ is cupping? (duality of course, when

the ambient manifold is an orientable, etc. 4-manifold)

 

[Bacle]

A quick followup, please:

It would seem that if there was no torsion, then, by the Universal Coefficient
Theorem of Homology, every Z/2 chain is the reduction of a Z-chain. Is this
a necessary and sufficient condition for an isomorphism between the chain groups?

Also: it seems like there is a relationship between simple-connectedness and 1-torsion
that also follows from the Universal Coefficient Theorem. Is this a fact?

Thanks.

 

[lavinia]

A quick followup, please:

It would seem that if there was no torsion, then, by the Universal Coefficient
Theorem of Homology, every Z/2 chain is the reduction of a Z-chain. Is this
a necessary and sufficient condition for an isomorphism between the chain groups?

Also: it seems like there is a relationship between simple-connectedness and 1-torsion
that also follows from the Universal Coefficient Theorem. Is this a fact?

Thanks.

If there is no torsion then then Bockstein sequence looks like

0 -> H_i(M:Z) -> H_i(M:Z) -> H_i(M:Z/2Z) -> 0

 

[lavinia]

Thanks, both.

I guess then that a characteristic surface would then be Poincare-

dual to the Stiefel-Whitney 2-class, right, i.e., a cocycle w^

with : w^\/ c = c\/c , where \/ is cupping? (duality of course, when

the ambient manifold is an orientable, etc. 4-manifold)

Your statement confuses me but ... I think it is true that Poincare duality works mod2 whether or not the manifold is orientable. This probably is because the tangent bundle of any manifold is orientable mod 2. So the 2'nd Stiefel-Whitney class is dual to a 2 chain - mod 2. If one triangulates the manifold then takes the first barycentric subdivision of the triangulation then this dual chain is just the mod 2 sum of all of the 2 simplices in the subdivision.

The duality map is - evaluating a mod 2 cocycle on the dual 2 chain is the same as cupping the cocycle with the 2'nd Stiefel-Whitney class and evaluating on the mod2 fundamental 4 cycle.