What are some examples of applications of integral quadratic forms in $n$ variables in algebraic topology?

Question

I'm reading the wiki page of qudratic forms. It simply seems curious to me what are some concrete examples of applications of integral quadratic forms in algebraic topology. I've searched a bit but a lot of the readings online are too involved. Hope to see some well-illustrated ones here. Thanks!

Answer 1

On a closed $4$-manifold $X$, define the intersection form $q_X:H^2(X)\otimes H^2(X) \to H^4(X) = \mathbb{Z}$ (coefficients are in $\mathbb{Z}$ throughout) by $q(\alpha, \beta) = \alpha \cap \beta$. (Such a form kills torsion, so it can be thought of a quadratic form over $\mathbb{Z}$.) If $X$ is simply-connected, then $q$ completely determines $X$ up to homotopy equivalence. In the opposite direction, any unimodular, symmetric bilinear form over $\mathbb{Z}$ is isomorphic to $q_X$ for some $X$. If $q$ is even, then $X$ is unique up to homeomorphism. If $q$ is odd, then there are two such manifolds up to homeomorphism. For smooth $4$-manifolds, homotopy equivalent manifolds are also homeomorphic. Thus for $q$ odd, at least one of the manifolds with intersection form $q$ must have no smooth structure. Generalizing this result, Donaldson proved that if $q_X$ is definite for $X$ (simply-connected and) smooth, then $q$ must be diagonalizable. In particular, there are quite a few $4$-manifolds that cannot be given a smooth structure. For manifolds that do have a smooth structure, it follows that any two such manifolds $X, X'$ are homeomorphic iff $q_X$ and $q_{X'}$ have the same rank, signature, and parity.

Answer 2

The paper

J.H.C. WHITEHEAD: A certain exact sequence. Ann. Math. 52 (1950), 51-110.

introduced a functor $\Gamma$ which is the ``universal quadratic functor" from Abelian groups to Abelian groups. Let $A$ be an Abelian group. Then $\Gamma(A)$ is the Abelian group with generators $\gamma a, a \in A$, and the following relations:

1. $ \gamma(-a)=\gamma(a), a\in A$

   2. if $\beta(a,b)=\gamma(a+b)-\gamma a -\gamma b, a,b\in A$, then $\beta : A \times A\to \Gamma(A)$ is biadditive.

This functor also occurs in the paper

R. BROWN and J.-L. LODAY, ``Van Kampen theorems for diagrams of spaces'', — Topology 26 (1987) 311-334.

This paper introduced a nonabelian tensor product $G \otimes H$ of groups $G,H$ which act on each other "compatibly", see this bibliography, so that in particular we have a tensor square $G \otimes G$ and a morphism of groups $\kappa: G \otimes G \to G$ induced by the commutator map $[\;,\;]: G \times G \to G$. The kernel of $\kappa$ is written $J_2(G)$ and is identified in the last paper as $\pi_3(SK(G,1))$. Further there is an exact sequence

$$H_3(G)\to \Gamma(G^{ab}) \to J_2(G) \to H_2(G) \to 0 . $$

I think there are more applications of $\Gamma$ in work of H,-J. Baues.