nLab MNinebrane

Idea

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

Definition

Let $Ninebrane(n)\coloneqq O(n)\langle 12\rangle$ be the $11$-connected cover in the Whitehead tower of the orthogonal group $O(n)$. (Following the notations $2-Orient(n)=BO(n) \langle 9\rangle$ and $2-Spin(n) =BO(n) \langle 10\rangle$, one could also write $2-String(n)=Ninebrane(n)$ to indicte its correspondence with the string group under Bott periodicity.) Through the canonical projection $p\colon Ninebrane(n)\twoheadrightarrow O(n)$, there is a pullback:

$$\gamma_{Ninebrane}^n \coloneqq p^*\gamma_\mathbb{R}^n \twoheadrightarrow BNinebrane(n).$$

Its Thom spectrum is the Ninebrane spectrum:

$$MNinebrane(n) \coloneqq\mathbf{Th}\left(\gamma_{Ninebrane}^n\right) =\Sigma^{\infty-n}Th\left(\gamma_{Ninebrane}^n\right).$$

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

$$\begin{aligned} MNinebrane(n) &=\Sigma^{\infty-n}Th\left(\gamma_{Ninebrane}^n\right) \cong\Sigma^{\infty-(n+1)}Th\left(\gamma_{Ninebrane}^n\oplus\underline{\mathbb{R}}\right) \\ &\rightarrow\Sigma^{\infty-(n+1)}Th\left(\gamma_{Ninebrane}^{n+1}\right) =MNinebrane(n+1). \end{aligned}$$

Its limit is denoted:

$$MNinebrane \coloneqq\lim_{n\rightarrow\infty}MNinebrane(n).$$

Connections

From the canonical projections $O(n)\langle 16\rangle\twoheadrightarrow Ninebrane(n)\twoheadrightarrow Fivebrane(n)$ and $O\langle 16\rangle\twoheadrightarrow Ninebrane\twoheadrightarrow Fivebrane$ , there are canonical spectrum morphisms:

$$MO(n)\langle 16\rangle\rightarrow MNinebrane(n)\rightarrow MFivebrane(n);$$

$$MO\langle 16\rangle\rightarrow MNinebrane\rightarrow MFivebrane.$$

Ninebrane bordism homology theory

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

$$\Omega_n^Ninebrane \cong\pi_n MNinebrane =\lim_{k\rightarrow\infty}\pi_k MNinebrane_{n+k}.$$

More general, MNinebrane defines a generalized homology theory (formally also denoted $\widetilde{MNinebrane}_*$) given by:

$$\Omega_n^Ninebrane(X) \coloneqq\pi_n^stab(X_+\wedge MNinebrane) \coloneqq\lim_{k\rightarrow\infty}\pi_{n+k}(X_+\wedge MNinebrane_k)$$

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

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

$$S^{n+k}\rightarrow X_+\wedge MNinebrane_k$$

representing the nine fundamental class $[X]\in\Omega_n^Ninebrane(X)$. Geometrically, it’s represented by the identity $id\colon X\rightarrow X$.

Ninebrane cobordism cohomology theory

MNinebrane also defines a generalized cohomology theory given by:

$$\widetilde{MNinebrane}^n(X) \coloneqq\lim_{k\rightarrow\infty}[\Sigma^k X,MNinebrane_{n+k}]$$

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