nLab spinᶜ bordism

Idea

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

Its definition is fully analogous to that of spin bordism. Similarily, there are spinᶜ bordism groups and the spinᶜ bordism ring:

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

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

Definition

Let $M$ and $N$ be $n$-dimensional spinᶜ manifolds with respective spinᶜ structures $\tau_M\colon M\rightarrow BSpin^c(n)$ and $\tau_N\colon N\rightarrow BSpin^c(n)$. A $n+1$-dimensional spinᶜ manifold $W$ with spinᶜ structure $\tau_W\colon W\rightarrow BSpin^c(n+1)$ 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 $k\colon Spin^c(n)\rightarrow Spin^c(n+1)$ is a spinᶜ 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

• 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]