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

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:


*

*$ \gamma(-a)=\gamma(a), a\in A$


*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. 
