Trace-Map Bordism

• Trace-map bordism is a collection of bordism-theoretic constructions that use canonical comparison, retraction, and transfer maps to relate various bordism theories.
• It employs natural transformations and splitting mechanisms—such as forgetful maps and modified counits in TQFTs—to address anomalies and ensure positive-definite pairings.
• Its diverse formulations connect singular, equivariant, and higher-categorical bordism with differential Anderson duality and cyclic phenomena, underpinning evaluation on bordism cycles.

Trace-map bordism denotes a cluster of bordism-theoretic constructions in which a bordism category or bordism homology theory carries a canonical comparison, retraction, evaluation, or higher-categorical trace. The terminology is not uniform across the literature. Several papers explicitly state that they do not define a map called “trace,” and instead isolate the closest analogue as a forgetful natural transformation, a transfer-like splitting, a Verdier projection, or a bordism-valued functional; by contrast, one later paper uses “trace-map bordism” directly for a modified counit in the Atiyah–Segal framework for Witten-type TQFTs (Friedman, 2013, Abouzaid et al., 2024, Gu, 26 Aug 2025). In this broader sense, trace-map bordism sits at the intersection of singular bordism, equivariant and orbifold bordism, cobordism-hypothesis trace theory, differential Anderson duality, and bordism-category models for transfer and cyclic phenomena.

1. Terminological status and conceptual range

The literature does not present a single, universally accepted object called a trace-map bordism. Instead, it exhibits several recurrent patterns. In singular bordism, the closest trace-like construction is often a canonical comparison map from a richer theory to a coarser one. In equivariant and orbifold settings, the relevant maps are frequently one-sided splittings or quotient-and-resolve morphisms. In higher-categorical settings, the trace itself is represented by bordisms with defects, and its moduli are computed by bordism hypotheses. In differential and anomaly-theoretic settings, the target is a dual bordism theory whose classes evaluate on bordism cycles (Friedman, 2013, Abouzaid et al., 2024, Steinebrunner, 2018, Han et al., 31 Oct 2025).

This dispersion of meaning is substantive rather than accidental. One-dimensional cobordism categories classify tracelike transformations $T_{x,C}:\mathrm{hom}_C(x,x)\to \mathrm{hom}_C(1,1)$ for dualisable $x$, while marked two-tori encode secondary traces for commuting endomorphisms of 2-dualizable objects (Steinebrunner, 2018, Ben-Zvi et al., 2013). By contrast, papers on pseudomanifold bordism, bordism versions of the $h$-principle, and assembly for bordism-invariant functors explicitly frame their central morphisms as comparison or localization maps rather than traces (Friedman, 2013, Sadykov, 2010, Levin et al., 5 Jun 2025). A precise encyclopedia treatment therefore treats trace-map bordism as a family of bordism-mediated trace mechanisms rather than a single definition.

2. Comparison morphisms in singular and formal bordism theories

In stratified pseudomanifold bordism, the central comparison is the forgetful natural transformation

$$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$

from stratified to unstratified bordism. For classes determined by local link conditions, this map is a natural isomorphism of homology theories, and at the level of absolute groups it specializes to

$$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$

The same work constructs an oriented bordism from a compact PL stratified pseudomanifold $X$ to its intrinsic stratification $X^*$, with underlying space $|I\times X|$, via the half-intrinsic suspension. It also proves that every unstratified bordism between $|X|$ and $|Z|$ can be endowed with a compatible stratification extending prescribed boundary stratifications. Together with the boundary map

$x$0

and the long exact sequence of a pair, these results provide the canonical comparison infrastructure most closely analogous to a trace in singular bordism (Friedman, 2013).

A parallel comparison appears in the bordism version of the $x$1-principle. For an open stable differential relation $x$2, Sadykov defines a natural transformation

$x$3

from the cohomology theory of genuine $x$4-solutions to that of stable formal solutions. It is induced by passage from an actual map to its normal stable formal differential data, and at the spectrum level it is represented by the map

$x$5

Under the paper’s b-principle hypotheses—always for $x$6, and for $x$7 provided each Morse function on a $x$8-manifold is a solution of $x$9—this comparison is an equivalence. The formal theory is identified with a Thom-spectrum model $h$0, so the map functions as a bordism-to-stable-homotopy comparison rather than a literal trace (Sadykov, 2010).

These two examples establish a common pattern: a trace-like bordism map often appears first as a natural transformation that forgets, stabilizes, or localizes structure while preserving the underlying bordism class.

3. Retractions, splittings, and transfer-like constructions

A different branch of trace-map bordism is represented by one-sided splittings. In bordism and resolution of singularities, the main map is the natural transformation

$h$1

which splits the inclusion of orbifold bordism into derived orbifold bordism. After passing to topological spaces by the representable-orbispace replacement functor $h$2, the paper obtains

$h$3

whose composite with the tautological map from manifold bordism is the identity. The construction is geometric on cycles. It first passes from derived orbifolds to orbifolds by choosing straightenings, perturbing to a regular Fukaya–Ono–Parker section, and resolving the singular zero loci. It then passes from orbifolds to manifolds by abelianization,

$h$4

followed by deorbification,

$h$5

so that the coarse space $h$6 is a smooth manifold (Abouzaid et al., 2024).

The same paper proves a finite-group splitting

$h$7

where a class $h$8 is sent to $h$9, and constructs a retraction

$$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$0

by sending a $$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$1-manifold to its quotient orbifold and then applying the orbifold-to-manifold splitting. It also proves the equivariant derived/geometric splitting

$$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$2

These splittings are natural transformations of homology theories, $$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$3-module linear, geometric on cycles, and explicitly one-sided: the reverse composite is not asserted to be the identity, and no spectrum-level splitting is proved in the paper (Abouzaid et al., 2024).

This branch of the subject is trace-like in the sense of retraction or transfer, not in the sense of categorical trace. The common structure is a canonical morphism from a singular, derived, or equivariant theory back to a smoother theory, together with a proof that the morphism splits an evident inclusion.

4. Cobordism-hypothesis traces and higher-categorical bordism

The most literal bordism-theoretic understanding of trace arises from the cobordism hypothesis. Hoyois, Scherotzke, and Sibilla define a tracelike transformation as a natural family of conjugation-invariant maps on endomorphisms of dualisable objects and compute the moduli spaces of such transformations using the one-dimensional cobordism hypothesis with singularities. In their formulation,

$$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$4

and the main calculation identifies

$$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$5

with a corresponding restricted formula involving $$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$6 and $$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$7. Ordinary trace extends uniquely to an $$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$8-categorical tracelike transformation up to a contractible space of choices, while the component of $$s:\Omega_*^{\mathcal E}(\cdot)\to \Omega_*^{|\mathcal E|}(\cdot),$$9 is $$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$0 for $$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$1 and $$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$2 for $$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$3 (Steinebrunner, 2018).

Ben-Zvi and Nadler lift this picture to dimension two. For a 2-dualizable object $$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$4 in a symmetric monoidal $$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$5-category, right dualizable endomorphisms $$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$6, and a commuting transformation

$$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$7

they define a secondary trace as an iterated trace and prove the canonical equivalence

$$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$8

Their proof is a Morse-theoretic analysis of a marked $$s:\Omega_n^{\mathcal C}\xrightarrow{\cong}\Omega_n^{|\mathcal C|}.$$9-torus and is explicitly presented as a concrete realization of the cobordism hypothesis with singularities. The two formulas correspond to different decompositions of the same marked bordism, and the shearing formula supplies the complementary $X$0-symmetry; together they encode the $X$1- and $X$2-moves of $X$3 (Ben-Zvi et al., 2013).

These papers give the strongest literal sense in which trace-map bordism is a bordism theory of traces: traces are values of field theories on defect-labeled bordisms, and trace identities are bordism invariance statements.

5. The explicit modified trace-map bordism in Witten-type TQFTs

One paper uses the expression “trace-map bordism” directly. In the Atiyah–Segal framework for 2d Witten-type TQFTs constructed from topological twists of mass-gapped theories, the usual counit

$X$4

is replaced by a modified counit with a background-particle insertion,

$X$5

where $X$6 and $X$7 is a distinguished background particle carrying the opposite anomalous axial charge. The associated pairing becomes

$X$8

This modification is introduced because the naive Atiyah–Segal trace can vanish on compact worldsheets by anomaly or charge-selection rules, and the paper identifies two symptoms: failure of the identity bordism, expressed by

$X$9

and apparent non-unitarity of the standard pairing in examples such as $X^*$0 (Gu, 26 Aug 2025).

The paper keeps multiplication $X^*$1, comultiplication $X^*$2, and unit $X^*$3 unchanged, but modifies the counit cap by inserting $X^*$4. It imposes

$X^*$5

and introduces an anti-background particle satisfying

$X^*$6

For the $X^*$7 NLSM, the insertion is

$X^*$8

In the examples discussed, this replacement turns the naive non-Hermitian or indefinite pairing into a diagonal positive-definite one when $X^*$9 is real. The construction is explicit, but its categorical status is deliberately limited: the paper conjectures self-consistency of the modified bordism category, does not provide a full sewing theorem, and treats only closed strings. Within the present literature, this is the clearest explicit object named trace-map bordism (Gu, 26 Aug 2025).

6. Bordism-valued traces, bordism duals, and evaluation on cycles

Trace-map bordism also appears when a trace takes values in a bordism theory or its dual. For two maps $|I\times X|$0 between closed manifolds of dimensions $|I\times X|$1 and $|I\times X|$2, Tsutaya defines the coincidence Reidemeister trace as

$|I\times X|$3

where $|I\times X|$4 is the homotopy equalizer and $|I\times X|$5 is the shriek map induced by the diagonal. The stable refinement is

$|I\times X|$6

and the paper proves that this coincides with Koschorke’s stabilized bordism invariant

$|I\times X|$7

For oriented $|I\times X|$8 and $|I\times X|$9, the Hurewicz image of $|X|$0 is precisely $|X|$1. In the range $|X|$2, vanishing of $|X|$3 is equivalent to deformability of $|X|$4 and $|X|$5 to coincidence-free maps. This is a direct instance of a homological trace as the shadow of a bordism-valued trace (Tsutaya, 2016).

A differential version of this dual perspective appears in the construction of differential models for the Anderson dual to twisted $|X|$6-bordism. A class in

$|X|$7

is represented by a pair $|X|$8, where $|X|$9 is an $|Z|$0-valued homomorphism on differential twisted bordism cycles and satisfies

$|Z|$1

The paper then defines the twisted anomaly map

$|Z|$2

whose functional component is given by a reduced eta-invariant together with explicit correction terms. This is not called a trace map, but the codomain is literally a theory of differential bordism characters, so the construction is trace-like in the precise sense of evaluation on bordism cycles (Han et al., 31 Oct 2025).

7. Transfer-adjacent bordism models and related operations

Several additional constructions clarify the geometric mechanics from which trace-map bordism often emerges. The circle transfer

$|Z|$3

admits a geometric reinterpretation as a morphism of cobordism categories: $|Z|$4 Here $|Z|$5 is the subcategory of $|Z|$6-dimensional cobordisms whose morphisms are unparametrized circles mapped to $|Z|$7, and the functor $|Z|$8 is induced by multiplication on $|Z|$9. Via Pontrjagin–Thom, the composite recovers the circle transfer. This places a standard transfer central to trace constructions into an explicit bordism-category framework (Giansiracusa, 2017).

A closely related development appears in the reduced one-dimensional bordism category $x$00, obtained by deleting closed circle components from composed bordisms. Its classifying space satisfies

$x$01

and for simply connected $x$02 of finite type the paper proves

$x$03

The reduction fiber sequence is driven by the circle transfer

$x$04

This is not a direct construction of the cyclotomic trace, but it gives a bordism model for the $x$05-side and records the trace-like effect of gluing as the creation and subsequent deletion of circle components (Steinebrunner, 2020).

Other constructions are trace-adjacent rather than traces proper. Symmetric squaring in singular bordism defines a natural map from $x$06 or $x$07 to a Čech bordism group of the quotient $x$08 after removing a neighborhood of the diagonal and dividing by the transposition involution. The resulting operation is norm-like or quotient-like, not literally a trace, but it provides a geometric template for bordism operations built from fixed-point singularities (Krempasky, 2012). In a categorical $x$09-theoretic direction, assembly for bordism-invariant functors of Poincaré categories is modeled by a Verdier projection

$x$10

whose evaluation under a bordism-invariant localizing invariant yields the assembly map; this again is not called a trace, but it is the closest analogue of a collapse from local bordism data to a global invariant (Levin et al., 5 Jun 2025). Finally, in the 3-dimensional setting of homology cobordisms of surfaces, Morita-type traces are packaged by the cocycle

$x$11

whose graded components recover the Morita traces of Johnson homomorphisms and relate them to the Magnus representation and the LMO homomorphism (Massuyeau et al., 2016).

Taken together, these constructions show that trace-map bordism is best understood as a structural theme: bordism supplies the universal geometry of closure, comparison, and evaluation, while “trace” may appear as an actual categorical trace, a retraction, a fixed-locus extraction, a bordism character, or a quotient map whose defect is itself measured by bordism.