Hi, All:


I am trying to see if there is a nice relation between two different definitions of quadratic form q; a topological definition $q\_T$, and an algebraic definition $q\_A$, and, if there is, how to go from one version to the other; either from version #1 below to version#2, or #2 to #1, or, even better, both ways.The two definitiions. Version #1 (defined below) is the topological version, and #2 (also defined below) is the algebraic version. My main interest here, actually, is to see if we can express the topological format as a homogeneous polynomial of degree 2 in {x,y} 
 This may be embarrasingly-simple for someone with a stronger Abstract-Algebra background than mine:


1)In algebra, a quadratic form $q\_A$ is a homogeneous polynomial P(x,y) of degree 2 in {x,y}; e.g., P(x,y)=$x^2+y^2$ 


2)In Topology/Homology, a quadratic form $q\_T$ is defined as a map $q\_T:Z\rightarrow Z\_2$, where Z is a finitely-dimensional $Z\_2$- vector space, associated with an intersection form ; more specifically, in my case of interest, Z is $H\_1(Sg,\mathbb Z\_2)$,where Sg is the genus-g surface, with the finite generating set (symplectic basis) {$x\_1,y\_1;x\_2,y\_2;....;x\_2g,y\_2g$} , with the intersection pairing $(x,y)\_2(mod2)$, defined by: $(x\_i,y\_j)\_2=1$ , if i=j, and 0 otherwise. Then
 $q\_T$ is defined to be a quadratic form , if it satisfies the identity:



```
 $q_T(x+y)=q_T(x)+q_T(y)+(x,y)_2$ , for all x,y in Z   (##)

```

In addition, we do know the values of $q\_T(x\_i$) and $q\_T(y\_i)$ , i.e., we know the value of the quadratic form in the basis elements.


So basically, I would like to be able to express my Topological $q\_T$ as a homogeneous, quadratic polynomial. The fact that (##) above looks a lot like the bilinear form associated with a(n) algebraic quadratic form makes me think that this may be possible, but the fact that we are working over $Z\_2$ makes me worry that, if possible, it may be difficult.



Are these Two Definitions of Quadratic Form (Algebraic, Topological) Related to Each Other?
There are some quick answers, but what you actually want is chapter V, section 1 in *Symmetric Bilinear Forms* by Milnor and Husemoller, pages 100-105, the section named "Homology Theory of Manifolds." They even include an appendix 1 called Quadratic Forms, pages 110-113. 


I put pages 100-105 at this [LINK](http://zakuski.utsa.edu/~jagy/milnor_homology.pdf)


