Rational Spin Bordism

• Rational Spin Bordism is the study of Spin-manifolds under the bordism equivalence relation, forming a rational ring that coincides with oriented bordism.
• Its ring structure is polynomial, generated by characteristic classes such as the Pontryagin classes derived from the cohomology of BSpin.
• Recent advancements provide explicit geometric bases using products of Hilbert schemes on K3 surfaces, linking hyperkähler geometry with classical bordism theory.

Rational Spin bordism is the study of Spin-manifolds under the equivalence relation of bordism, where two closed manifolds with Spin-structure on their stable normal bundles are declared bordant if they together bound a compact Spin-manifold. The associated Spin–bordism group in degree $k$, denoted $\Omega_k^{\mathrm{Spin}}$, is tensored with the rational numbers %%%%2%%%% to form the rational Spin–bordism ring, $\Omega_*^{\mathrm{Spin}} \otimes \mathbb{Q}$. Recent advancements provide explicit geometric bases for this ring using Hilbert schemes of points on K3 surfaces, relying on deep results connecting hyperkähler geometry, characteristic classes, and the Milnor genus (Buchanan et al., 26 Jan 2026).

1. Formal Definition and Relation to Oriented Bordism

Let $\mathrm{MSpin}$ be the Thom spectrum for Spin-structures on stable normal bundles. The Spin–bordism ring is identified as $$\Omega_*^{\mathrm{Spin}} \cong \pi_*(\mathrm{MSpin})$$. Passing to the rationalized theory yields $$\Omega_*^{\mathrm{Spin}} \otimes \mathbb{Q} \cong \pi_*(\mathrm{MSpin}) \otimes \mathbb{Q}$$. This object is graded by the real dimension of the representative manifolds.

A foundational result (Anderson–Brown–Peterson) asserts $$\Omega_*^{\mathrm{Spin}} \otimes \mathbb{Q} \cong \Omega_*^{\mathrm{SO}} \otimes \mathbb{Q}$$—rational Spin–bordism coincides with rational oriented bordism. In particular, the forgetful map from rational symplectic bordism to rational Spin–bordism is a graded ring isomorphism (Theorem 3.7), yielding the canonical identifications: $$\Omega_*^{\mathrm{Sp}} \otimes \mathbb{Q} \cong \Omega_*^{\mathrm{Spin}} \otimes \mathbb{Q} \cong \Omega_*^{\mathrm{SO}} \otimes \mathbb{Q}.$$ This equivalence implies that, up to torsion, the additional structure given by Spin does not alter the rational classification of bordism classes.

2. Ring Structure and Characteristic Classes

The rational Spin–bordism ring inherits a polynomial structure from the cohomology ring of $B\mathrm{Spin}$. Thom–Hirzebruch theory and characteristic class computations show

$$\Omega_*^{\mathrm{Spin}}\otimes\mathbb{Q} \cong H^*(B\mathrm{Spin};\mathbb{Q}),$$

and classical results (Milnor–Stasheff [MS74]) provide

$$H^*(B\mathrm{Spin};\mathbb{Q}) \cong \mathbb{Q}[p_1,p_2,\ldots], \qquad |p_i| = 4i,$$

with $p_i$ the $i$-th Pontryagin class. Alternatively, $$\Omega_*^{\mathrm{Sp}} \otimes \mathbb{Q} \cong \mathbb{Q}[q_1, q_2, \ldots]$$, $|q_i| = 4i$, where symplectic Pontryagin classes $q_i$ relate to $p_i$ by $q_i \mapsto a_i p_i$, $a_i \in \mathbb{Q}^\times$ (Proposition 2.6). It follows that $\Omega_*^{\mathrm{Spin}}\otimes\mathbb{Q}$ is a free commutative polynomial algebra on a generator in each degree congruent to $0$ modulo $4$.

3. Explicit Construction of Basis Elements

The principal innovation is the presentation of explicit generators for $\Omega_{4n}^{\mathrm{Spin}} \otimes \mathbb{Q}$ as products of Hilbert schemes of K3 surfaces: $$B_r^{4n} = \left\{ \prod_{i=1}^k K3^{[n_i]} \mid (n_1, \ldots, n_k)\ \text{partition of}\ n \right\},$$ where $K3^{[m]}$ is the Hilbert scheme of $m$ points on a fixed complex K3 surface. Each $K3^{[m]}$ is hyperkähler (Beauville [Bea83]), which entails a canonical Spin-structure via its holonomy (Proposition 3.3, Corollary 3.5).

Every element of $B_r^{4n}$ is a manifold of real dimension $4n$, and the collection spans all monomial degrees. In the pure Spin case, tori and other factors play no role—basis elements are exactly unordered products of these Hilbert schemes without further geometric summands.

4. Milnor Genus, OSV Theorem, and Linear Independence

The completeness and independence of these generators follows from the theorem of Oberdieck–Song–Voisin (OSV) [OSV22], which identifies a "Milnor genus" linear functional $M: \Omega_{4n}^U \otimes \mathbb{Q} \to \mathbb{Q}$ via a formal power series: $$Q(t) = \frac{t}{\epsilon(e^t - 1)},\quad \epsilon^2 = -2.$$ For a K3 surface $X$, with $X^{[n]}$ its $n$-point Hilbert scheme and $c_1(\mathcal{O}(1))$ the first Chern class, OSV show that

$$M(X^{[n]}) = \left\langle \exp(\epsilon\, c_1(\mathcal{O}(1))), [X^{[n]}] \right\rangle$$

provides, after evaluating against all monomials in Chern numbers indexed by partitions of $n$, a non-singular $P(n)\times P(n)$ matrix, where $P(n)$ counts the partitions of $n$. This invertibility guarantees that the span of bordism classes $[X^{[n]}]$ in complex cobordism is full in the "even" subring, and a fortiori in rational Spin–bordism (Lemma 4.1, Lemma 4.5).

Dimension counting shows $B_r^{4n}$ contains $P(n)$ elements, matching $$\dim_{\mathbb{Q}} \Omega_{4n}^{\mathrm{Spin}} \otimes \mathbb{Q}$$ (Corollary 3.9), establishing both spanning and independence.

5. Examples in Low Dimensions and Basis Table

The structure in lower degrees is entirely explicit. The table below summarizes the low-dimensional rational Spin–bordism groups and their bases:

Degree $k$	$\Omega_k^{\mathrm{Spin}} \otimes \mathbb{Q}$	Basis $B_r^k$
$0$	$\mathbb{Q}$	$[\mathrm{pt}]$
$1,2,3$	$0$	—
$4$	$\mathbb{Q}$	$\{K3\}$
$8$	$\mathbb{Q}^2$	$\{ K3^{[2]},\ K3 \times K3 \}$
$12$	$\mathbb{Q}^3$	$$\{ K3^{[3]},\ K3^{[2]}\times K3, K3\times K3\times K3 \}$$

For each $4n$, $B_r^{4n}$ consists of all unordered products $\prod_i K3^{[n_i]}$ as $n$ runs over all partitions, with each such product sitting in real dimension $4n$.

6. Structural Results and Open Questions

The main structural result (Theorem 1.2) states that the forgetful map $$\Omega_*^{\mathrm{Sp}} \otimes \mathbb{Q} \rightarrow \Omega_*^{\mathrm{Spin}} \otimes \mathbb{Q}$$ is an isomorphism of graded rings. Furthermore, one can select generators corresponding to explicit hyperkähler manifolds—in the Spin case, these are precisely products of Hilbert schemes of K3 surfaces in dimension $4n$ and nothing else.

Open problems include the status of these classes in integral Spin bordism and over $\mathbb{Z}[1/2]$, and the potential for hyperkähler geometry to control or detect torsion phenomena. There are analogous questions for six related cobordism theories: symplectic (Sp), complex and quaternionic refinements (Sp$^c$, Sp$^h$), and Spin$^c$, Spin$^h$ versions.

7. Significance and Further Directions

The identification of canonical geometric generators for rational Spin–bordism, grounded in the geometry of hyperkähler varieties and the theory of Hilbert schemes on K3 surfaces, significantly sharpens the understanding of the landscape of differentiable manifolds up to Spin bordism when rational structure is imposed. This framework unifies the treatment of complex, oriented, and symplectic cobordism in the rational regime and connects modern developments in hyperkähler and holomorphic symplectic geometry to classical questions in homotopy theory and characteristic classes. The methods employed are indicative of new possibilities for interpreting and computing in related generalized cohomology theories and raise substantive questions regarding the relationship between integral and rational classifications, the role of torsion, and the reach of geometric constructions in bordism groups (Buchanan et al., 26 Jan 2026).