nLab MSpinᶜ

Idea

The cobordism theory for manifolds equipped with (stable) spinᶜ structure.

Properties

Atiyah-Bott-Shapiro orientation

(…)

see at Atiyah-Bott-Shapiro orientation

(…)

Relation to complex K-theory

$M Spin^c$ is related to KU in a variant of the Conner-Floyd isomorphism, via the Atiyah-Bott-Shapiro orientation (Hopkins-Hovey 92, Thm. 1)

Spinᶜ bordism homology theory

According to Thom's theorem, there is an isomorphism to spinᶜ bordism groups:

$$\Omega_n^{Spin^\mathrm{c}} \cong\pi_n MSpin^\mathrm{c} =\lim_{k\rightarrow\infty}\pi_k MSpin_{n+k}^\mathrm{c}.$$

More general, MSpinᶜ defines a generalized homology theory (formally also denoted $\widetilde{MSpin^\mathrm{c}}_*$) given by:

$$\Omega_n^{Spin^\mathrm{c}}(X) \coloneqq\pi_n^stab(X_+\wedge MSpin^\mathrm{c}) \coloneqq\lim_{k\rightarrow\infty}\pi_{n+k}(X_+\wedge MSpin^\mathrm{c}_k)$$

for all topological spaces $X$ with the disjoint union $X_+\coloneqq X+\{*\}$. Since $\{*\}_+\cong S^0$ is the neutral element of the wedge product, one has $\Omega_n^{Spin^\mathrm{c}}=\Omega_n^{Spin^\mathrm{c}}(*)$. Geometrically, $\Omega_n^{Spin^\mathrm{c}}(X)$ can also be described by $n$-dimensional spinᶜ manifolds representing cycles and $n+1$-dimensional spinᶜ bordisms? representing homologous cycles, which are mapped continuous into $X$. For a detailed explanation see spinᶜ bordism.

A $n$-dimensional spinᶜ manifold $X$ has a spinᶜ fundamental class $[X]\in\Omega_n^{Spin^\mathrm{c}}(X)$. Let $i\colon X\hookrightarrow\mathbb{R}^{n+k}\hookrightarrow S^{n+k}$ be an embedding (which always exists due to the Whitney embedding theorem), then its Pontrjagin-Thom collapse map is:

$$S^{n+k}\rightarrow X_+\wedge Th(N_i X)$$

with the normal bundle $N_i X\coloneqq TS^{n+k}/i^*TX$. Since the spinᶜ structure of $X$ transfers over to its stable normal bundle? ($N_i X$ for $k\rightarrow\infty$), postcomposition yields the map:

$$S^{n+k}\rightarrow X_+\wedge MSpin^\mathrm{c}_k,$$

which represents the spinᶜ fundamental class $[X]\in\Omega_n^{Spin^\mathrm{c}}(X)$. Geometrically, it’s represented by the identity $id\colon X\rightarrow X$.

Spinᶜ cobordism cohomology theory

MSpinᶜ also defines a generalized cohomology theory given by:

$$\widetilde{MSpin^\mathrm{c}}^n(X) \coloneqq\lim_{k\rightarrow\infty}[\Sigma^k X,MSpin^\mathrm{c}_{n+k}]$$

for all topological spaces $X$. It can also be described geometrically.´with spinᶜ structures.

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

• Michael Hopkins, Mark Hovey, Spin cobordism determines real K-theory, Mathematische Zeitschrift volume 210, pages 181–196 (1992) (doi:10.1007/BF02571790, pdf)