Spectral Viterbo isomorphism: complex-oriented versus framed

Abstract: The Viterbo isomorphism relates the symplectic cohomology of a cotangent bundle to the homology of the free loop space of its base. We lift this to a relation of modules over (1) the complex bordism spectrum MU and (2) the sphere spectrum $\mathbb{S}$. In particular, by a result of Porcelli and the present author [BP26], it is not the case that, in general, the $\mathbb{S}$-level statement recovers the MU-level statement after base-change -- even in the case the base is spin.

• The paper proves that the framed Floer homotopy type of T*Q is equivalent to the suspension spectrum of the free loop space, Σ∞₊LQ, for every base Q without orientability or spin assumptions.
• The paper identifies the canonical MU-Floer homotopy type with a Thom spectrum over LQ twisted by an explicitly constructed MU-local system from Cauchy–Riemann index theory, recovering the integral Viterbo isomorphism with local coefficients.
• The results show that the MU refinement contains genuinely new information not obtained by base change from the sphere spectrum, while non-oriented cases generally require breaking S¹ loop-rotation symmetry.

The Viterbo isomorphism identifies the symplectic cohomology of a cotangent bundle $T^*Q$ with the homology of the free loop space $LQ$ of its base. In Floer homotopy theory, this statement admits refinements at the level of spectra: the framed Floer homotopy type of $T^*Q$, associated to the standard stable $R$-polarization, should recover $\Sigma^\infty_+ LQ$ as an object of the stable $\infty$-category of spectra, while the canonical complex-oriented Floer homotopy type $F^{\rm MU}$ — which exists for any graded Liouville manifold because the moduli spaces of Floer trajectories carry canonical stable complex structures — should be identified with some Thom spectrum over $LQ$. This paper by Kenneth Blakey establishes both statements in full generality, without orientability or spin hypotheses on the base (2608.20289).

Background and motivation

Classically, the isomorphism takes the form

$SH^{-*}(T^*Q;\mathbb{Z})\cong H_*(LQ;\eta),$

where $\eta$ is a local system on $LQ$0 that is trivial when $LQ$1 is spin. The statement was first proven with $LQ$2-coefficients (Viterbo; Abbondandolo–Schwarz; Salamon–Weber), and Kragh observed that it fails integrally when $LQ$3 is non-spin, a phenomenon verified by Seidel's computation of $LQ$4. A complete integral proof appears in Abouzaid's monograph.

On the spectral side, prior work established special cases: Cohen proved the spin case of the $LQ$5-level statement, Côté–Kartal handled the stably framed base in the $LQ$6-equivariant setting, and Asplund–Deshmukh–Pieloch treated the open-string analogue. The general framed case existed only in an unpublished note of the author. The impetus for the present work comes from Blakey–Porcelli, who showed that the canonical MU-Floer homotopy type of $LQ$7, for odd $LQ$8, cannot be obtained by base-changing any spectrum along $LQ$9. Since $T^*Q$0 is spin, this yields a strong conclusion: even in the spin case, the $T^*Q$1-level Viterbo isomorphism does not recover the MU-level statement after base-change. Consequently, the MU-Viterbo isomorphism must involve genuinely new data — an MU-local system on $T^*Q$2 that does not lift to $T^*Q$3.

Main results

Let $T^*Q$4 denote the canonical MU-Floer homotopy type of $T^*Q$5 with its standard Liouville structure. The main theorem states:

MU-level: There exists an MU-local system $T^*Q$6 such that

$T^*Q$7

recovering $T^*Q$8.

Sphere level: For the framed Floer homotopy type $T^*Q$9 associated to the standard stable $R$0-polarization,

$R$1

recovering $R$2.

Here $R$3 is the local system that is trivial on loops whose pullback tangent bundle is orientable and $R$4 on those where it is not, and $R$5 is constructed purely through index theory of Cauchy–Riemann operators. The local system $R$6 itself is defined via spin trivializations of $R$7, the orientation local system, and a degree shift depending on orientability of $R$8.

A key structural point is that the construction of $R$9 deliberately breaks the $\Sigma^\infty_+ LQ$0-symmetry of loop rotation: the author fixes the line $\Sigma^\infty_+ LQ$1 on the cylinder to obtain canonical identifications of index bundles in the non-oriented case. This is consistent with the observation that, for non-oriented $\Sigma^\infty_+ LQ$2, the equivalence of framed $\Sigma^\infty_+ LQ$3-structures underlying the BV algebra structures holds only with $\Sigma^\infty_+ LQ$4-gradings, so one should not expect the spectral isomorphisms to respect $\Sigma^\infty_+ LQ$5-equivariance in general.

Method

The proof follows the framework of structured flow categories of Abouzaid–Blumberg, using the equivalences $\Sigma^\infty_+ LQ$6, $\Sigma^\infty_+ LQ$7, and their spherical variants classifying bordism of MU-oriented resp. framed manifolds. The strategy proceeds through finite-dimensional approximations $\Sigma^\infty_+ LQ$8 of the loop space, Morse flow categories $\Sigma^\infty_+ LQ$9 lifting to $\infty$0, and Floer flow categories $\infty$1 built from admissible Hamiltonian data linear at infinity.

Three ingredients are combined:

1. Twisted orientations on hybrid moduli spaces. The Viterbo bimodule is defined via half-cylinder moduli spaces $\infty$2 evaluated into $\infty$3 and intersected with Morse stable manifolds. A central lemma produces a canonical trivialization of the MU-local system $\infty$4 twisted by $\infty$5, obtained by gluing the cap operator $\infty$6 to the linearized Cauchy–Riemann operator and passing to the doubled operator, whose index is complex-linear and hence canonically equivalent to MU.
2. Compatibility with directed systems. Homotopies between the composite bimodules involving continuation maps and inclusion maps are constructed via flow 2-simplices, with twisted orientations extending over Gromov compactifications and restricting correctly to codimension-one boundary strata.
3. Descent from spherical to genuine structures. The map induced on $\infty$7-modules agrees on integral homology with Abouzaid's Viterbo isomorphism; since all spectra involved are bounded below, a spectral Whitehead theorem upgrades it to a homotopy equivalence. Results of Porcelli–Smith then lift the spherical complex-oriented data to genuine complex-oriented flow categories, and conservativity of base-change along $\infty$8 (which admits a retraction) yields the MU-level identification.

The framed case is parallel but simpler: the twisted stable framing $\infty$9 follows from the symmetry-breaking lemma identifying index bundles of glued operators after fixing the marked point on the cylinder, and the determinant lines of the abstract caps reproduce precisely the local system $F^{\rm MU}$0, so that base-change along $F^{\rm MU}$1 recovers Abouzaid's chain-level isomorphism.

Limitations and open questions

The paper concedes several points. The construction of the MU-local system $F^{\rm MU}$2 is purely analytic, via Cauchy–Riemann operators; whether it has a purely homotopy-theoretic origin is posed as a conjecture. Specifically, using real Bott periodicity and the canonical nullhomotopy of $F^{\rm MU}$3, one obtains a candidate local system $F^{\rm MU}$4 built from the stable tangent class $F^{\rm MU}$5, and the conjecture asserts that the Thom spectra of $F^{\rm MU}$6 and $F^{\rm MU}$7 are equivalent. Proving this equivariantly would require showing that the left-hand side carries an appropriate $F^{\rm MU}$8-action, since the map $F^{\rm MU}$9 breaks $LQ$0-symmetry.

Second, the symmetry-breaking is forced by the non-oriented generality; in the oriented case an alternative lemma preserving $LQ$1-symmetry applies. This leaves open the question posed explicitly in the paper: what is the statement of the MUP-Viterbo isomorphism respecting the framed $LQ$2-structures, i.e., what is the MUP-local system on $LQ$3 that does not break $LQ$4-symmetry? Relatedly, Rezchikov's cyclotomic structures on $LQ$5-Floer homotopy types suggest further equivariant refinements not addressed here.

Finally, the Fredholm analysis suppresses some technical points — weighted Sobolev spaces are needed to handle kernel issues in the asymptotic operators defining $LQ$6 — and the transversality arguments rely on the machinery of Large's thesis and Porcelli–Smith rather than being reproduced in full.

Conclusion

This paper completes the spectral refinement of the Viterbo isomorphism in two directions simultaneously: it proves the general framed statement $LQ$7 without orientability assumptions, and it formulates and proves the correct MU-level analogue, identifying the canonical MU-Floer homotopy type with a Thom spectrum over $LQ$8 twisted by an explicitly constructed, index-theoretic MU-local system. The comparison with Blakey–Porcelli shows decisively that these two levels are inequivalent even for spin bases, with the obstruction concentrated in the non-liftability of the MU-local system to the sphere spectrum. The remaining questions concern the homotopy-theoretic nature of the twisting local system and its compatibility with loop rotation.