A bounded lattice on the test sphere
Fix an odd integer \(d \geq 3\). This chapter compares the continuous theory \(E_r = W_r(\T(-))\) with the semialgebraic theory \(F_r = W_r(\Pcat(-))\) on spheres, shows the rational degree-\(d\) group is the line spanned by the orientation class, and bounds the integral image inside a fixed lattice \(L_d^{-1}\Z\bar u_d\). The denominator \(L_d\) depends only on \(d\): it is fixed before any action, test map, or character partition enters. Two further properties connect the lattice to the action: degree and integral localization away from a contractible set.
Comparison and the lattice
For every integer \(r\), the comparison \(F_r(\pt) \to E_r(\pt)\) is an isomorphism, and both equal \(\Z\) for \(r \equiv 0 \pmod 4\) and \(0\) otherwise. In degree zero the isomorphism is signature. On \(S^0\) both theories are the direct sum of two point theories, and the map to the point is addition.
Proof sketch. Both point categories are exactly the bounded derived category of finite-dimensional real vector spaces. A degree \(r\) form splits into cohomology pairings; off-diagonal terms are hyperbolic pairs, odd \(r\) leaves nothing, \(r = 2k\) with \(k\) odd is alternating hence hyperbolic, and \(r = 4m\) is a real symmetric form whose class is its signature. Signature respects neutral relations because a lagrangian triangle exhibits the image as self-perpendicular in middle degree.
Dependencies
In a commutative diagram of exact rows \(A_1 \to \cdots \to A_5\) over \(B_1 \to \cdots \to B_5\), if the kernels and cokernels of \(f_1, f_2, f_4, f_5\) are annihilated by \(q\), then those of \(f_3\) are annihilated by \(q^3\). In a commutative square with both horizontal kernels and cokernels annihilated by \(2\), if those of \(f\) are annihilated by \(q\), then those of \(g\) are annihilated by \(4q\) (the cokernel even by \(2q\)).
Proof sketch. Two elementary diagram chases: lift through exact rows, clearing denominators at each of the three bounded stages for the five-term chase, and through the square for the second.
The kernel bound for \(f_1\) is unused by the chase and is kept to match the source's symmetric statement. The cokernel bound for \(f_5\) is likewise unused and kept for the same reason. In the square, the kernel bound for \(i\) is unused and kept for the same reason.
Dependencies
For every \(j \geq 0\) there is a nonnegative integer \(e_j\), independent of \(r\), such that the kernel and cokernel of \(F_r(S^j) \to E_r(S^j)\) are annihilated by \(2^{e_j}\) for every integer \(r\).
Proof sketch. Induction on \(j\) with \(e_0 = 0\). For the pole cover \(X = U \cup V\) with \(I = U \cap V \simeq S^{j-1}\), homotopy invariance makes the comparisons on \(U, V\) isomorphisms and bounds those on \(I\). A five-term chase through the localization rows bounds the comparison on relative groups; the square chase with the exponent-two dense maps transfers it across the cover; a final five-term chase bounds the middle comparison. The exponents compose as \(e \mapsto 3e \mapsto 3e+2 \mapsto 9e+6\), so one may take \(e_j = 3(9^j-1)/4\), but only existence is used.
Dependencies
For either \(G_r = F_r\) or \(G_r = E_r\), and \(j \geq 0\), \(G_r(S^j) \otimes \Q \simeq (G_r(\pt) \otimes \Q) \oplus (G_{r-j}(\pt) \otimes \Q)\). In particular, for odd \(d\) both \(F_d(S^d) \otimes \Q\) and \(E_d(S^d) \otimes \Q\) are one-dimensional rational vector spaces.
Proof sketch. Split off the point summand. Localization identifies the relative group of the pole cover with the reduced group, and the other localization sequence identifies its boundary as an isomorphism; the dense map between the two relative groups is a rational isomorphism, giving \(\widetilde G_r(S^j) \otimes \Q \simeq \widetilde G_{r-1}(S^{j-1}) \otimes \Q\). At \((r,j) = (d,d)\) the summands are \(0\) and \(\Q\) by the point calculation.
Dependencies
Every finite collection of objects of \(\Pcat(S^d)\) has finite-dimensional locally constant cohomology on a common dense open subset of \(S^d\). On a sufficiently small oriented ball there, a degree-\(d\) form has an integral signature on its degree-zero cohomology, and for any finite Witt relation one ball serves all objects and lagrangian diagrams simultaneously, so signatures respect the relation. Consequently \(u_{F,d}\) is nontorsion and \(F_d(S^d) \otimes \Q = \Q\bar u_{F,d}\) with integral image exactly \(\Z\bar u_{F,d}\).
Proof sketch. Hardt triviality plus cell refinement gives a dense open set where each generator has finite locally constant cohomology; the property passes through cones since extensions of finite local systems split on small balls. On such a ball the complex is a finite constant complex, duality becomes ordinary derived linear duality, and a lagrangian triangle gives an ordinary lagrangian, hence signature zero. Clearing denominators against the ball signature shows every rational coefficient is integral.
Dependencies
Let \(d \geq 3\) be odd with orientation class \(u_d\). There is a positive power of \(2\), \(L_d = 2^{e_d}\), such that \(E_d(S^d) \otimes \Q = \Q\bar u_d\) and the integral image satisfies \(\im(E_d(S^d) \to E_d(S^d) \otimes \Q) \subseteq L_d^{-1}\Z\bar u_d\). This denominator is fixed by the sphere dimension and comparison, independent of every later action, test map, character partition, prime, and exponent.
Proof sketch. Bounded comparison makes the comparison a rational isomorphism, identifying \(\bar u_d\) with the nonzero rational orientation class. Since \(L_d\) annihilates the comparison cokernel in degree \(d\), every integral class times \(L_d\) lifts to the semialgebraic side, where its rational image is an integral multiple of the orientation class.
Interface to the action
Let \(d \geq 3\) be odd. If \(f : S^d \to S^d\) is continuous of degree \(s \in \Z\), then \(f_*\bar u_d = s\bar u_d\) in \(E_d(S^d) \otimes \Q\); hence \(f_*\) acts on the whole rational group by \(s\). More generally, if \(N\) is a connected closed oriented triangulable \(d\)-manifold and \(f : N \to S^d\) has degree \(s\), its pushed orientation form has rational class \(s\bar u_d\).
Proof sketch. Simplicially approximate \(f\) after subdivision on semialgebraic models; homotopy invariance identifies \(f_*\) and \(g_*\) on the continuous theory. Over a ball in a top-dimensional target simplex, the map is finitely many homeomorphic sheets; the pushed pairing is diagonal with \(\pm 1\) entries by orientation signs, and its signature is the degree by the local-degree formula. Clearing denominators against a signature ball inside the sheet ball determines the rational coefficient; the comparison transfers it to \(E\), where \(\bar u_d\) spans.
Let \(d \geq 3\) be odd, \(X\) any test space for \(\T\), and \(U \subset X\) a nonempty contractible open subset. The quotient functor induces an integral isomorphism \(E_d(X) \xrightarrow{\sim} W_d(\T(X)/\T(U))\), natural for continuous \(g : X \to X'\) with \(g(U) \subset U'\) for contractible \(U'\).
Proof sketch. Homotopy invariance plus the point calculation gives \(E_d(U) = 0\), and \(E_{d-1}(U) \to E_{d-1}(X)\) is split injective via projection to the point (in both odd congruence classes of \(d\) mod 4). The localization sequence then collapses to the asserted integral isomorphism; no rationalization or splitting of quotient idempotents is involved. Naturality follows from the natural localization sequence since \(g_*\) preserves support subcategories.