nLab pinᶜ bordism

Idea

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

Pinᶜ bordism groups

Bahri-Gilkey gave two computations of pinᶜ bordism groups: one using homotopy theory, and one using analytic techniques. The first several groups are:

• $\Omega_0^{Pin^\mathrm{c}}\cong\mathbb{Z}_2$
• $\Omega_1^{Pin^\mathrm{c}}\cong 1$
• $\Omega_2^{Pin^\mathrm{c}}\cong\mathbb{Z}_4$
• $\Omega_3^{Pin^\mathrm{c}}\cong 1$
• $\Omega_4^{Pin^\mathrm{c}}\cong\mathbb{Z}_8\oplus\mathbb{Z}_2$
• $\Omega_5^{Pin^\mathrm{c}}\cong 1$
• $\Omega_6^{Pin^\mathrm{c}}\cong\mathbb{Z}_16\oplus\mathbb{Z}_4$
• $\Omega_7^{Pin^\mathrm{c}}\cong 1$

(Bahri-Gilkey 1987, Theorem 2)

Related entries

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

The general computation of pinᶜ bordism in all degrees:

• Anthony Bahri and Peter Gilkey. $\operatorname{Pin}^c$ Cobordism and Equivariant $\operatorname{Spin}^c$ Cobordism of Cyclic 2-Groups. Proceedings of the American Mathematical Society, vol. 99, no. 2, 1987. doi:10.2307/2046645.

• Anthony Bahri and Peter Gilkey. The eta invariant, $\operatorname{Pin}^c$ Cobordism and Equivariant $\operatorname{Spin}^c$ Cobordism of Cyclic 2-Groups. Pacific Journal of

  Mathematics, vol. 28, no. 1, 1987.

Some explicit low-degree computations:

• Arun Debray: Fun with $E(1)$-modules: A computation of pinᶜ bordism (2020) [pdf, pdf]