nLab spinʰ bordism

Idea

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

Its definition is fully analogous to that of spin bordism. Similarily, there are spinʰ bordism groups:

$$\Omega_n^{Spin^\mathrm{h}} \coloneqq\pi_n MSpin^\mathrm{h} =\lim_{k\rightarrow\infty}\pi_{n+k} MSpin^\mathrm{h}_k;$$

Unlike for spin or spinᶜ structure, there is not in general a spinʰ structure on the direct sum of two spinʰ vector bundles. Thus the direct product of spinʰ manifolds in general doesn’t admit a spinʰ structure. See e.g. Albanese-Milivojević, Example 2.8.

For many B-bordism theories, a B-structure on the direct sum of two B-structured vector bundles is used to define a commutative ring spectrum structure on the Thom spectrum MB. The previous paragraph implies this argument does not work for MSpinʰ – and indeed, MSpinʰ does not even have a homotopy ring spectrum structure. If it did, the $\pi_*(\mathbb{S})$-module structure on $\pi_*(\mathrm{MSpin}^h)$ would refine to a $\pi_*(\mathbb{S})$-algebra structure, but if $\eta\in\pi_1(\mathbb{S})\cong\mathbb{Z}/2$ denotes the nonzero element, then $\eta\cdot 1 = 0\in\pi_1(\mathrm{MSpin}^h)$ and $\eta\cdot[\mathbb{HP}^1]\ne 0$ in $\pi_5(\mathrm{MSpin}^h)$, which follows from the spinʰ bordism computations of Freed-Hopkins.

However, the direct sum of the spinʰ bordism groups

$$\Omega^{Spin^\mathrm{h}} \coloneqq\bigoplus_{n\in\mathbb{N}}\Omega_n^Spin^{\mathrm{h}}.$$

is a module over the spin bordism ring, and likewise MSpinʰ is an MSpin-module spectrum.

Definition

Let $M$ and $N$ be $n$-dimensional spinʰ manifolds with respective spinʰ structures $\tau_M\colon M\rightarrow BSpin^h(n)$ and $\tau_N\colon N\rightarrow BSpin^h(n)$. A $n+1$-dimensional spinʰ manifold $W$ with spinʰ structure $\tau_W\colon W\rightarrow BSpin^h(n+1)$ 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 $k\colon Spin^h(n)\rightarrow Spin^h(n+1)$ is a spinʰ bordism between $M$ and $N$. It is fully denoted by $(W,M,N,i,j)$, but usually $W$ is sufficient from context.

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

• Daniel S. Freed and Michael J. Hopkins, Reflection positivity and invertible topological phases, Geometry and Topology 25 (2021). arXiv:1604.06527, [doi:10.2140/gt.2021.25.1165]

• Michael Albanese and Aleksandar Milivojević, Spinʰ and further generalisations of spin, Journal of Geometry and Physics 164 (2021). arXiv:2008.04934, [doi:10.1016/j.geomphys.2021.104174]

• Keith Mills, The Structure of the Spin^h Bordism Spectrum (2023) [arXiv:2306.17709v2]

• 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]