Intersection Form With Coeffs. in Z/2

[Bacle]

Hi, Everyone:

Just wondering if anyone knew about how to work with the intersection form with
coefficients in Z/2. I only know this is in relation to Wu's vector, tho I don't know
what Wu's vector is.

I was also hoping to know if the intersection form for (4n+2)-manifolds is also symmetric,
i.e., as in the case for 4n-manifolds; just wonder if there is a (-1)ⁿ⁺¹ or something
that may change signs.

Thanks for any Comments, Refs.

 

[quasar987]

I believe the intersection form in dimension 4n+2 is is antisymetric .

Because the intersection form is the bilinear form on H^{2n+1}(M) given by taking the cup product followed by evaluation on the fundamental class µ: (a,b) --> <a u b, µ>. And cup product is commutative in the graded sense: a u b = (-1)^|a||b| b u a = (-1)^(2n+1)² b u a = -b u a.

Also, from what I understand (may be wrong), by Poincaré duality, the intersection form of an compact n-manifold M can also be seen as a Z-valued bilinear form on H_{n/2}(M) (homology this time). Given two homology classes [L1], [L2], it is possible to find a representant L1, L2 of those classes such that L1 and L2 are dimension n/2 submanifolds of M which intersect transversally, and hence in finite number of points since M is compact. The intersection form then counts this number of intersection points. I'm guessing that working mod 2, the intersection only tells you about the parity of the number of intersection points.

 

[lavinia]

I believe the intersection form in dimension 4n+2 is is antisymetric .

Because the intersection form is the bilinear form on H^{2n+1}(M) given by taking the cup product followed by evaluation on the fundamental class µ: (a,b) --> <a u b, µ>. And cup product is commutative in the graded sense: a u b = (-1)^|a||b| b u a = (-1)^(2n+1)² b u a = -b u a.

Also, from what I understand (may be wrong), by Poincaré duality, the intersection form of an compact n-manifold M can also be seen as a Z-valued bilinear form on H_{n/2}(M) (homology this time). Given two homology classes [L1], [L2], it is possible to find a representant L1, L2 of those classes such that L1 and L2 are dimension n/2 submanifolds of M which intersect transversally, and hence in finite number of points since M is compact. The intersection form then counts this number of intersection points. I'm guessing that working mod 2, the intersection only tells you about the parity of the number of intersection points.

This seems correct. I guess over Z/2 you can define the intersection form for unorientable manifolds.

But I have a question. If the manifold is orientable there still may be non-orientable half dimension manifolds. Over Z I guess these do not represent homology classes so they are not a problem?

 

[quasar987]

I don't understand the nature of your question. What is the problem that you see regarding unorientable dimension n\2 submanifolds?

 

[lavinia]

I don't understand the nature of your question. What is the problem that you see regarding unorientable dimension n\2 submanifolds?

well they have intersection just like anything else but they can only be counted mod2. So if they represented homology classes something would be wrong. Sorry I guess I answered my own question. they are not homology classes and do not enter into the intersection product.

 

[quasar987]

Oh I see. So in post #2, I should have said

"Given two homology classes [L1], [L2], it is possible to find orientable representatives L1, L2 of those classes such that L1 and L2 are dimension n/2 submanifolds of M which intersect transversally, and hence in finite number of points since M is compact. The intersection form then counts this (signed) number of intersection points. I'm guessing that working mod 2, the intersection only tells you about the parity of the number of intersection points. "