nLab ninebrane bordism

Idea

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

Definition

Let $M$ and $N$ be $n$-dimensional ninebrane manifolds with respective ninebrane structures $\tau_M\colon M\rightarrow BNinebrane(n)$ and $\tau_N\colon N\rightarrow BNinebrane(n)$. A $n+1$-dimensional ninebrane manifold $W$ with ninebrane structure $\tau_W\colon W\rightarrow BNinebrane(n+1)$ 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 $k\colon Ninebrane(n)\rightarrow Ninebrane(n+1)$ is a ninebrane bordism between $M$ and $N$. It is fully denoted by $(W,M,N,i,j)$, but usually $W$ is sufficient from context.

Ninebrane bordism groups

Under the equivalence relation of ninebrane bordism, all $n$-dimensional closed ninebrane manifolds form the ninebrane bordism group $\Omega_n^Ninebrane$, which as the disjoint union as composition, the empty manifold as neutral element and the inversion of orientation as inversion. According to Thom's theorem, ninebrane bordism groups are exactly the stable homotopy groups of the Thom spectrum MNinebrane:

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

Since $BNinebrane=BO\langle 16\rangle$ is $15$-connected, the first fifteen ninebrane bordism groups ($0\leq n\leq 14$) coincide with the framed bordism groups?:

• $\Omega_0^Ninebrane\cong\Omega_0^fr\cong\mathbb{Z}$
• $\Omega_1^Ninebrane\cong\Omega_1^fr\cong\mathbb{Z}_2$
• $\Omega_2^Ninebrane\cong\Omega_2^fr\cong\mathbb{Z}_2$
• $\Omega_3^Ninebrane\cong\Omega_3^fr\cong\mathbb{Z}_24$
• $\Omega_4^Ninebrane\cong\Omega_4^fr\cong 1$
• $\Omega_5^Ninebrane\cong\Omega_5^fr\cong 1$
• $\Omega_6^Ninebrane\cong\Omega_6^fr\cong\mathbb{Z}_2$
• $\Omega_7^Ninebrane\cong\Omega_7^fr\cong\mathbb{Z}_240$

Ninebrane bordism ring

All ninebrane bordism groups in a direct sum form the ninebrane bordism ring:

$$\Omega^\mathrm{Ninebrane} \coloneqq\bigoplus_{n\in\mathbb{N}}\Omega_n^\mathrm{Ninebrane},$$

which has the cartesian product as additional composition and the singleton as an additional neutral element.

Properties

Proposition

Every $n$-dimensional ninebrane manifold is ninebrane bordant to a $\min\left\{15,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected ninebrane manifold, equivalently meaning that every ninebrane bordism homology class in $\Omega_n^Ninebrane$ can be represented by such a ninebrane manifold. (For $n\geq 31$, the result stabilizes at a 15-connected ninebrane manifold.)

(Botvinnik & Labbi 14, Lem. 3.2 (1))

Proposition

For $n$-dimensional $\min\left\{15,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected ninebrane manifolds $M$ and $N$, a ninebrane bordism $W\colon M\rightsquigarrow N$ exists with $M\hookrightarrow W$ also $\min\left\{15,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected.

(Botvinnik & Labbi 14, Lem. 3.2 (2))

Proposition

If a $n$-dimensional $k$-connected compact ninebrane manifold $M$ with $k\leq 14$ and $n\geq 2k+3$ is ninebrane bordant to another compact ninebrane manifold $N$, then $M$ can be obtained from $N$ by surgery of codimension at least $k+2$.

(Botvinnik & Labbi 14, Prop. 3.4)

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

• Hisham Sati, Ninebrane structures (arXiv:1405.7686)

• Boris Botvinnik, Mohammed Labbi, Highly connected manifolds of positive $p$-curvature, Transactions of the AMS, Trans. Amer. Math. Soc. 366 (2014), 3405-3424 [arXiv:1201.1849, doi:10.1090/S0002-9947-2014-05939-4]