Virtual Cobordism: Theory and Applications

• Virtual cobordism is a framework that generalizes classical cobordism to virtual settings in algebraic geometry and topology.
• It utilizes methods like virtual pullbacks and refined Gysin morphisms to define and compute invariants in moduli spaces and enumerative geometry.
• In knot theory, virtual cobordism extends to virtual knots and links, enabling classification via virtual linking numbers and new concordance invariants.

Virtual cobordism encompasses a suite of mathematical frameworks generalizing classical cobordism concepts to a range of “virtual” contexts: derived algebraic geometry, singular moduli, and knot/link theories with virtual crossings. Virtual cobordism classes serve as fundamental tools for enumerative invariants in moduli problems, for the classification of virtual string links, and for the extension of intersection-theoretic operations to singular and derived settings. This article surveys core structures, universal properties, and applications of virtual cobordism, with emphasis on algebraic-geometric, homotopical, and knot-theoretic aspects.

1. Virtual Cobordism in Derived Algebraic Geometry

The paradigm of virtual cobordism in algebraic geometry was developed to handle moduli spaces which are not smooth, but carry perfect obstruction theories, as in enumerative problems of Gromov–Witten, Donaldson–Thomas, and stable pair theories. The main construction, as outlined by Annala–Yokura, proceeds in the functorial context of quasi-projective derived schemes over a Noetherian base $A$ of finite Krull-dimension.

A universal bivariant theory $\mathbb{M}^A_*(X\to Y)$ is constructed out of projective morphisms $V\to X$ equipped with quasi-smooth (derived lci) maps $V\to Y$ of prescribed virtual dimension. Imposing double-point relations yields the universal precobordism $\mathcal{P}^A_*(X\to Y)$, which satisfies foundational axioms such as the section axiom and the formal group law (FGL) for Chern classes.

To specialize to algebraic cobordism $\Omega^A_*(X\to Y)$, one further quotients by simple normal crossing relations, ensuring the theory is universal among stably oriented, additive, commutative bivariant theories satisfying Section, FGL, and snc conditions. This realization coincides with both classical and derived algebraic cobordism, and the construction is compatible with all expected pushforward, pullback, and product operations (Annala, 2020).

2. Virtual Pullbacks and the Structure of Virtual Classes

Given a quasi-smooth map or a perfect obstruction theory, one constructs virtual pullbacks in cobordism. If $i: Z \hookrightarrow X$ is a quasi-smooth closed immersion, there exists a morphism, described as a composition of pullbacks and push-forwards along derived deformation spaces and conormal bundles, denoted

$i^!_{\mathrm{vir}}: \Omega_*(X)\to \Omega_{*-c}(Z)$

where $c$ is the virtual codimension. This virtual Gysin morphism coincides with the usual one for classical lci morphisms and is compatible with compositions, base-change in derived Cartesian squares, and push-pull identities (Annala, 2020).

For moduli spaces $M$ with perfect obstruction theory—such as spaces of stable pairs—one obtains a virtual cobordism class $\mathbb{M}^A_*(X\to Y)$0, which is functorial, and whose virtual fundamental class is canonically constructed in the algebraic cobordism ring. The class is characterized by virtual tangent bundle data and is realized via deformation to the normal cone, refined Gysin pullback, and quotienting by obstruction theory (Shen, 2014).

3. Universal Characterizations and Operational Theories

Virtual cobordism admits a universal property: given any bivariant theory $\mathbb{M}^A_*(X\to Y)$1 on derived schemes over $\mathbb{M}^A_*(X\to Y)$2 satisfying Section, FGL, and (optionally) snc axioms, there exists a unique orientation-preserving Grothendieck transformation from $\mathbb{M}^A_*(X\to Y)$3 (or $\mathbb{M}^A_*(X\to Y)$4) to $\mathbb{M}^A_*(X\to Y)$5. This universality enables canonical identifications between seemingly disparate definitions—annular, operational, and derived formulations are equivalent over any $\mathbb{M}^A_*(X\to Y)$6, not just fields (Annala, 2020).

For fields of characteristic zero, operational algebraic cobordism as constructed by González–Karu is isomorphic to the universal bivariant theory $\mathbb{M}^A_*(X\to Y)$7, under which every class acts via pushforward and refined pullback on all test spaces over $\mathbb{M}^A_*(X\to Y)$8. The transformation is compatible with all bivariant operations and the theory coincides with all prior constructions (Annala, 2020).

4. Applications: Moduli, Enumerative Invariants, and Partition Functions

In the theory of moduli of stable pairs on smooth 3-folds, the virtual cobordism class $\mathbb{M}^A_*(X\to Y)$9 encodes both classical enumerative information and all Chern-number data of the virtual tangent bundle. The partition function

$V\to X$0

lies in a formal power series ring over the cobordism group, and specializations to other oriented Borel–Moore homology theories recover invariants such as virtual Chern numbers. Rationality and functional equation properties of these partition functions (in $V\to X$1) generalize known properties from Calabi-Yau geometry, and have been established in toric cases via virtual localization (Shen, 2014).

In Gromov–Witten theory, quantization and twisting results for virtual tangent bundles allow the construction of complex cobordism-valued invariants, relating them to $V\to X$2-theoretic Gromov–Witten invariants by explicit quantized symplectic transformations. This structure is compatible with the universality of cobordism and enables specializations to variants such as Hirzebruch $V\to X$3-theory (Huq-Kuruvilla, 2021).

5. Virtual Cobordism in Knot Theory: Classification and Invariants

In knot theory, virtual cobordism generalizes classical link cobordism to virtual knots, links, and string links, allowing for virtual crossings and non-planar diagram moves. For $V\to X$4-strand virtual string links, Gaudreau established that virtual cobordism classes are classified by pairwise virtual linking numbers, yielding a bijection

$V\to X$5

Given any vector of linking numbers, there exists a unique (up to cobordism) standard diagram achieving them, and any two diagrams with the same data are virtually cobordant. Unwelded string links admit the same classification, while the welded one-component case is trivial under concordance (Gaudreau, 2019).

For virtual knots, virtual cobordism is generated by virtual isotopy, birth/death of unknotted components, and oriented saddle moves. The genus of a cobordism surface is controlled by counts of these moves. Numerous invariants have been developed: the affine index polynomial (a concordance invariant for knots/links), Turaev’s graded genus (sensitive to sliceness), and recently, group-valued invariants sensitive to generalized “$V\to X$6-moves” not detected by prior polynomial invariants (Kauffman, 2018, Manturov, 2022).

6. Comparison with Classical and Derived Settings

Virtual cobordism bridges classical intersection theory, derived approaches, and quantum invariants. In the algebraic-geometric context, virtual cobordism classes recover classical lci pullbacks when the relevant maps are classical, and all base change, composition, and pushforward compatibilities are inherited. The operational and functorial framework ensures that virtual cobordism is flexible enough to admit universal comparison isomorphisms among derived, classical, and operational theories (Annala, 2020).

In knot theory, virtual cobordism elucidates distinctions between classical and virtual sliceness, demonstrates the subtlety of concordance and pass-moves, and exhibits new phenomena not present in the classical case, such as non-trivial sliceness obstructions detectable by group-valued invariants.

7. Open Problems and Ongoing Directions

Key open directions in virtual cobordism include full classification of virtual knots and links up to concordance, computation and structure of the virtual knot concordance group, refinement and generalization of invariants (notably, extension to quantum or Floer-type theories and higher categories), and further elucidation of the interplay between moduli-theoretic virtual classes and topological approaches to cobordism. New invariants sensitive to moves such as the $V\to X$7-move indicate potential for detecting previously invisible structure in virtual categories, both algebraic and topological (Manturov, 2022).

The universality and operational realization of virtual cobordism ensure broad applicability across areas as diverse as enumerative geometry, quantum field theory, and low-dimensional topology.