Q-Formers: Extended Quadratic Forms

• Q-Formers are extended quadratic forms that generalize classical forms over integers by incorporating a quadratic form parameter Q with specific algebraic homomorphisms.
• Their structure, defined as a triple (X, λ, μ), satisfies identities linking bilinear forms and quadratic refinements, thereby handling both symmetric and anti-symmetric cases.
• Q-Formers are crucial in classifying manifolds and computing Witt groups, impacting algebraic topology, L-theory, and algebraic surgery through clear invariants.

A $Q$-former, or extended quadratic form with values in a quadratic form parameter $Q$, generalizes the classical notion of a quadratic form over the integers. It is rooted in the algebraic study of form parameters, developed to facilitate classification problems in topology and algebra, notably in the work of Wall on $(q{-}1)$-connected $2q$-manifolds. The systematic theory of $Q$-formers encodes bilinear and quadratic structures with enhanced flexibility and has implications for Witt group theory, bilinear analysis, L-theory, and the algebraic classification of manifolds (Crowley et al., 2024).

1. Quadratic Form Parameters and Their Structure

A quadratic form parameter over $\mathbb{Z}$ ("form parameter", often abbreviated as $Q$) is defined as a triple $Q = (Q_e, h, p)$, where $Q_e$ is a finitely generated abelian group, and $h: Q_e \to \mathbb{Z}$, $p: \mathbb{Z} \to Q_e$ are group homomorphisms satisfying: $$h \circ p \circ h = 2 h,\quad p \circ h \circ p = 2p$$ The value $E_Q \in \{ \pm 1 \}$ (the "symmetry" of $Q$) is determined via $h p - \mathrm{Id} = E_Q\,\mathrm{Id}$. If $E_Q = +1$, $Q$ is symmetric; if $E_Q = -1$, it is anti-symmetric. This structure is foundational for defining $Q$-formers, and governs their symmetry and algebraic invariants.

2. Definition and Properties of $Q$-Formers

Given a form parameter $Q = (Q_e, h, p)$ and a finitely generated (free) abelian group $X$, an extended quadratic form, or $Q$-former, is a triple $(X, \lambda, \mu)$ with:

• $\lambda: X \times X \to \mathbb{Z}$ a bilinear form satisfying $\lambda(ax, by) = ab \lambda(x, y)$ and $\lambda(x, y) = E_Q \lambda(y, x)$,
• $\mu: X \to Q_e$ a map (the "quadratic refinement") such that, for all $x, y \in X$,

$$\mu(x + y) = \mu(x) + \mu(y) + p(\lambda(x, y)), \quad h(\mu(x)) = \lambda(x, x)$$

This framework subsumes various classical settings; for instance, for the standard $Q=(\mathbb{Z},\,\mathrm{Id},\,0)$, a $Q$-former encodes a symmetric quadratic form. The formalism of $Q$-formers allows precise handling of both symmetric and anti-symmetric settings.

3. Classification of Form Parameters

Form parameters decompose into indecomposables, which are (up to isomorphism):

• Symmetric parameters ($E_Q=+1$):
  • $$Q^+ = (\mathbb{Z} \oplus \mathbb{Z},\,h(n, m) = n,\,p(1) = (0, 0))$$
  • $$Q^+_k = (\mathbb{Z} \oplus \mathbb{Z}_{2^k},\,h(n, [m]) = n \bmod 2,\,p(1) = (1, 1)),\; k \ge 1$$
• Anti-symmetric parameters ($E_Q=-1$):
  • $Q^- = (0 \to \mathbb{Z} \to 0)$
  • $$Q^-_k = (\mathbb{Z}_{2^k},\,h = 0,\,p(1) \ne 0),\; k \ge 2$$

Every form parameter splits (canonically, up to isomorphism) as $Q \cong Q_\text{indec} \oplus G$, with $Q_\text{indec}$ one of the indecomposables and $G$ a free abelian group. Form parameters organize into the category $\mathbf{FP}$ of form parameters, with morphisms $a: P \to Q$ group homomorphisms $a: P_e \to Q_e$ such that $h_Q \circ a = h_P$, $p_Q \circ a = p_P$. The category $\mathbf{FP}$ splits as $\mathbf{FP}_+ \sqcup \mathbf{FP}_-$ for the symmetric and anti-symmetric cases, with explicit equivalences: $\mathbf{FP}_+ \simeq (\text{abelian groups %%%%47%%%% with } v: A \to \mathbb{Z}_2) \equiv Ab / \mathbb{Z}_2$

$\mathbf{FP}_- \simeq (\text{abelian groups %%%%48%%%% under } \mathbb{Z}_2) \equiv \mathbb{Z}_2 / Ab$

4. Witt Groups of $Q$-Formers

For a form parameter $Q$, the Witt group $W_0(Q)$ is defined as the group of Witt classes of nonsingular $Q$-formers modulo metabolic forms. Explicitly, a $Q$-former $(X, \lambda, \mu)$ is nonsingular if the pairing $\lambda$ induces an isomorphism $X \to X^*$. It is metabolic if there exists a Lagrangian $L \subset X$ (half-rank, with $\lambda|_{L \times L} = 0$, $\mu(L) = 0$). Two nonsingular $Q$-formers are Witt-equivalent if they become isometric after orthogonal sum with metabolic forms. $W_0(Q)$ is computed as a functor $FP \to \mathrm{Ab}$, with orthogonal sum as the group operation.

Key results for indecomposable $Q$ are:

• $Q = Q^+\!:\quad W_0(Q^+) \cong 8\mathbb{Z}$ (signature invariant)
• $$Q = Q^+_k\!:\quad W_0(Q^+_k) \cong \mathbb{Z} \oplus \mathbb{Z}/2^{k-1}$$ (generators: signature and Milnor-type $p_k$)
• $Q = Q^-:\quad W_0(Q^-) \cong 0$
• $Q = Q^-_k:\quad W_0(Q^-_k) \cong \mathbb{Z}/2$ (Arf-invariant).

For a split parameter $P = Q \oplus G$, there exists a splitting $W_0(P) \cong W_0(Q) \oplus W_\text{red}(Q; G)$, where $W_\text{red}(Q; G) \cong G \otimes_Q Q_e$. This quadratic tensor product (in the sense of Baues) unifies various classical functors, such as $S^2(G)$, $P^2(G)$, $T(G)$, and $A_1(G)$.

5. Alternative Constructions and Exact Sequences

For symmetric $Q$, the extended symmetrisation $es: Q \to Q^+ \oplus S_Q$ induces an embedding $$W_0(es): W_0(Q) \hookrightarrow \mathbb{Z} \oplus T(S_Q)$$, with image described explicitly via subgroups generated by signature and kernel data. For anti-symmetric $Q$, the extended quadratic lift $eql: Q^- \oplus Q_e \to Q$ yields a surjection $$W_0(eql): \mathbb{Z}_2 \oplus A_1(Q_e) \twoheadrightarrow W_0(Q)$$, with kernel generated by explicit relations on the quadratic data. Exact sequences describe the splitting of quadratic tensor products:

• Symmetric: $$0 \to S^2(G) \to G \otimes_Q Q_e \to G \otimes S_Q \to 0$$
• Anti-symmetric: $$0 \to G \otimes Q_e \to G \otimes_Q Q_e \to A_1(G) \to 0$$

These constructions clarify image and kernel structures for canonical maps between Witt groups and related functors.

6. Applications and Connections

$Q$-formers have central importance in the classification of manifolds, notably in Wall’s work on simply-connected topology via $Q$-forms, and in algebraic surgery theory and Grothendieck-Witt theory. The described formalism unifies treatments of classical bilinear/quadratic form theory, $L$-groups of $\mathbb{Z}$, and algebraic invariants like the Arf-invariant, signature modulo $8$, and Milnor invariants. The Baues quadratic tensor product and derived exact sequences recover and generalize classical homological and cohomological operations, enabling the computation of manifold invariants from explicit $Q$-formers.

7. Summary and Examples

The theory provides a complete classification of quadratic form parameters over $\mathbb{Z}$, canonical splittings into indecomposables and free parts, and explicit computation of Witt groups for all cases. Invariants include the signature, Arf-invariant, Milnor-type operations, and constructions of canonical subgroups and quotients that describe the images and kernels of map between Witt groups. For special $Q$, the Witt groups recover the classical $L_n(\mathbb{Z})$. The splitting $W_0(P) \cong W_0(Q) \oplus (G \otimes_Q Q_e)$ for parameter $P = Q \oplus G$ is realized through natural exact sequences and the structure of the Baues quadratic tensor product, providing detailed invariants central to both algebraic and geometric topology (Crowley et al., 2024).