nLab spin bordism

Idea

A spin bordism is a B-bordism for the tangential structure ((B,f)-structure) being the spin structure. Its bordism homology theory and cobordism cohomology theory are described by the Thom spectrum MSpin.

Definition

Let $M$ and $N$ be $n$-dimensional spin manifolds with respective spin structures $\tau_M\colon M\rightarrow BSpin(n)$ and $\tau_N\colon N\rightarrow BSpin(n)$. A $n+1$-dimensional spin manifold $W$ with spin structure $\tau_W\colon W\rightarrow BSpin(n+1)$ together with inclusions $i\colon M\hookrightarrow\partial W$ and $j\colon N\hookrightarrow\partial W$ so that:

$$\partial W =i(M)+j(N);$$

$$\mathcal{B}k\circ\tau_M =\tau_W\circ i;$$

$$\mathcal{B}k\circ\tau_N =\tau_W\circ j$$

with the canonical inclusion $k\colon Spin(n)\rightarrow Spin(n+1)$ is a spin bordism between $M$ and $N$. It is fully denoted by $(W,M,N,i,j)$, but usually $W$ is sufficient from context.

Spin bordism groups

Under the equivalence relation of spin bordism, all $n$-dimensional closed spin manifolds form the spin bordism group $\Omega_n^Spin$, which has the disjoint union as composition, the empty manifold as neutral element and the inversion of orientation as inversion. According to Thom's theorem, spin bordism groups are exactly the stable homotopy groups of the Thom spectrum MSpin:

$$\Omega_n^Spin \cong\pi_n MSpin =\lim_{k\rightarrow\infty}\pi_{n+k}MSpin_k.$$

Since $BSpin = BO\langle 4\rangle$ is $3$-connected, the first three spin bordism groups ($0\leq n\leq 2$) coincide with the framed bordism groups?:

• $\Omega_0^Spin \cong \Omega_0^fr\cong\mathbb{Z}$, generated by the point space,

• $\Omega_1^Spin \cong \Omega_1^fr\cong\mathbb{Z}_2$,

• $\Omega_2^Spin \cong \Omega_2^fr\cong\mathbb{Z}_2$.

Further spin bordism groups include:

• $\Omega_3^Spin\cong 1$,

• $\Omega_4^Spin \cong \mathbb{Z}$, generated by the Kummer surface,

• $\Omega_5^Spin \cong \Omega_6^Spin\cong\Omega_7^Spin\cong 1$,

• $\Omega_8^Spin \cong \mathbb{Z}^2$, generated by the quaternionic projective plane $\mathbb{H}P^2$ and an 8-manifold obtained from the Kummer surface,

• $\Omega_11^Spin \cong 1$, which plays a role in the discussion of 11D supergravity (“M-theory”).

Here is a table of the first 100 spin (co)bordism groups:

(from Buchanan-McKean 2026)

Spin bordism ring

All spin bordism groups in a direct sum form the spin bordism ring:

$$\Omega^Spin \coloneqq\bigoplus_{n\in\mathbb{N}}\Omega_n^Spin,$$

which has the cartesian product as additional composition and the singleton as an additional neutral element.

Properties

Proposition

Every $n$-dimensional spin manifold is spin bordant to a $\min\left\{3,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected spin manifold, equivalently meaning that every spin bordism homology class in $\Omega_n^Spin$ can be represented by such a spin manifold. (For $n\geq 7$, the result stabilizes at a 3-connected spin manifold.)

(Botvinnik & Labbi 14, Lem. 3.2 (1))

Proposition

For $n$-dimensional $\min\left\{3,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected spin manifolds $M$ and $N$, a spin bordism $W\colon M\rightsquigarrow N$ exists with $M\hookrightarrow W$ also $\min\left\{3,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected.

(Botvinnik & Labbi 14, Lem. 3.2 (2))

Proposition

If a $n$-dimensional $k$-connected compact spin manifold $M$ with $k\leq 2$ and $n\geq 2k+3$ is spin bordant to another compact spin manifold $N$, then $M$ can be obtained from $N$ by surgery of codimension at least $k+2$.

(Botvinnik & Labbi 14, Prop. 3.4)

Related concepts

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MSpin^c, MSpin^h MString, MFivebrane, M2-Orient, M2-Spin, MNinebrane (see also pin⁻ bordism, pin⁺ bordism, pinᶜ bordism, spin bordism, spinᶜ bordism, spinʰ bordism, string bordism, fivebrane bordism, 2-oriented bordism, 2-spin bordism, ninebrane bordism)

• MU, MSU, MΩΩSU(n)

• MP, MR

• MSp

• MTO, MTSO

• relative bordism theories: MOFr, MUFr, MSUFr

• equivariant bordism theory: equivariant MFr, equivariant MO, equivariant MU

• global equivariant bordism theory: global equivariant mO, global equivariant mU

• algebraic: algebraic cobordism

References

About general $BO\langle\ell\rangle$ bordisms:

• Boris Botvinnik, Mohammed Labbi, Highly connected manifolds of positive $p$-curvature, Transactions of the AMS, Trans. Amer. Math. Soc. 366 (2014), 3405-3424 [arXiv:1201.1849, doi:10.1090/S0002-9947-2014-05939-4]

• Jonathan Buchanan, Stephen McKean: KSp-characteristic classes determine Spin cobordism, Algebr. Geom. Topol. 26 (2026) 485-551 [arXiv:2312.08209, doi:10.2140/agt.2026.26.485]

See also:

• Manifold Atlas, Spin bordism