nLab MSpinʰ

Idea

A spinʰ spectrum is the Thom spectrum of the universal vector bundle over a spinʰ group. Their limit over the infinite spinʰ group is of particular interest since its generalized homology theory describes spinʰ bordisms.

Definition

Let $Spin^\mathrm{h}(n)\coloneqq (Spin(n)\times Sp(1))/\mathbb{Z}_2$ be the spinʰ group. Through the canonical projection $p\colon Spin^\mathrm{h}(n)\twoheadrightarrow O(n)$, there is a pullback:

$$\gamma_{Spin^\mathrm{h}}^n \coloneqq p^*\gamma_\mathbb{R}^n \twoheadrightarrow BSpin^\mathrm{h}(n).$$

Its Thom spectrum is the Spinʰ spectrum:

$$MSpin^\mathrm{h}(n) \coloneqq\mathbf{Th}\left(\gamma_{Spin^\mathrm{h}}^n\right) =\Sigma^{\infty-n}Th\left(\gamma_{Spin^\mathrm{h}}^n\right).$$

The desuspension assures the invariance under the Whitney sum with trivial bundles, so $MSpin^\mathrm{h}(n)=\mathbf{Th}\left(\gamma_{Spin^\mathrm{h}}^n\oplus\underline{\mathbb{R}^m}\right)$. It also assures that the canonical inclusion $i\colon Spin^\mathrm{h}(n)\rightarrow Spin^\mathrm{h}(n+1)$, which pulls back to a canonical vector bundle homomorphism $\gamma_{Spin^\mathrm{h}}^n\oplus\underline{\mathbb{R}}=i^*\gamma_{Spin^\mathrm{h}}^{n+1}\rightarrow\gamma_{Spin^\mathrm{h}}^{n+1}$, induces a spectrum homomorphism:

$$MSpin^\mathrm{h}(n) =\Sigma^{\infty-n}Th\left(\gamma_{Spin^\mathrm{h}}^n\right) \cong\Sigma^{\infty-(n+1)}Th\left(\gamma_{Spin^\mathrm{h}}^n\oplus\underline{\mathbb{R}}\right) \rightarrow\Sigma^{\infty-(n+1)}Th\left(\gamma_{Spin^\mathrm{h}}^{n+1}\right) =MSpin^\mathrm{h}(n+1).$$

Its limit is denoted:

$$MSpin^\mathrm{h} \coloneqq\lim_{n\rightarrow\infty}MSpin^\mathrm{h}(n).$$

Connections

From the canonical projections $Spin(n)\hookrightarrow Spin^\mathrm{c}(n)\hookrightarrow Spin^\mathrm{h}(n)$ and $Spin\hookrightarrow Spin^\mathrm{c} \hookrightarrow Spin^\mathrm{h}$, there are canonical spectrum morphisms:

$$MSpin(n)\rightarrow MSpin^\mathrm{c}(n)\rightarrow MSpin^\mathrm{h}(n);$$

$$MSpin\rightarrow MSpin^\mathrm{c}\rightarrow MSpin^\mathrm{h}.$$

Examples

Due to the isomophisms $Spin^\mathrm{h}(1)\cong Sp(1)\cong SU(2)$ and $Spin^\mathrm{h}(2)\cong U(2)$ there are isomorphisms:

$$MSpin^\mathrm{h}(1) \cong MSp(1) \cong MSU(2);$$

$$MSpin^\mathrm{h}(2) \cong MU(2).$$

Spinʰ bordism homology theory

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

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

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

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

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 $\Omega_n^{Spin^\mathrm{h}}=\Omega_n^{Spin^\mathrm{h}}(*)$. Geometrically, $\Omega_n^{Spin^\mathrm{h}}(X)$ can also be described by $n$-dimensional spinʰ manifolds representing cycles and $n+1$-dimensional spinʰ bordisms? representing homologous cycles, which are mapped continuous into $X$. For a detailed explanation see spinʰ bordism.

A $n$-dimensional spinʰ manifold $X$ has a spinʰ fundamental class $[X]\in\Omega_n^{Spin^\mathrm{h}}(X)$. 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 spinʰ structure of $X$ transfers over to its stable normal bundle? ($N_i X$ for $k\rightarrow\infty$), postcomposition yields the map:

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

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

Spinʰ cobordism cohomology theory

MSpinʰ also defines a generalized cohomology theory given by:

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

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

Properties

Proposition

There is a 2-local homotopy equivalence of spectra:

$$MSpin\wedge\Sigma^{-3}MSO_3\rightarrow MSpin^\mathrm{h}$$

(Mills 23, Lem. 3.1)

Proposition

There is an isomorphism of $MSpin$ module spectra:

$$MSpin\wedge\Sigma^{-3}MSO_3\rightarrow MSpin^\mathrm{h}$$

(Mills 23, Crl. 3.2)

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

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