nLab fivebrane bordism

Idea

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

Definition

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

Fivebrane bordism groups

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

$$\Omega_n^Fivebrane \cong\pi_n MFivebrane =\lim_{k\rightarrow\infty}\pi_{n+k}MFivebrane_k.$$

Since $BFivebrane=BO\langle 9\rangle$ is $8$-connected, the first eight fivebrane bordism groups ($0\leq n\leq 7$) coincide with the framed bordism groups?:

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

Fivebrane bordism ring

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

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

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

Properties

Proposition

Every $n$-dimensional fivebrane manifold is fivebrane bordant to a $\min\left\{8,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected fivebrane manifold, equivalently meaning that every fivebrane bordism homology class in $\Omega_n^Fivebrane$ can be represented by such a fivebrane manifold. (For $n\geq 17$, the result stabilizes at a 8-connected fivebrane manifold.)

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

Proposition

For $n$-dimensional $\min\left\{8,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected fivebrane manifolds $M$ and $N$, a fivebrane bordism $W\colon M\rightsquigarrow N$ exists with $M\hookrightarrow W$ also $\min\left\{8,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected.

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

Proposition

If a $n$-dimensional $k$-connected compact fivebrane manifold $M$ with $k\leq 7$ and $n\geq 2k+3$ is fivebrane bordant to another compact fivebrane manifold $N$, then $M$ can be obtained from $N$ by surgery of codimension at least $k+2$.

(Botvinnik & Labbi 14, Prop. 3.4 & Crl. 3.6)

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, Urs Schreiber, Jim Stasheff, Fivebrane structures, Reviews in Mathematical Physics, 21 10 (2009) 1197-1240 [arXiv:0805.0564, doi:10.1142/S0129055X09003840]

• Christopher Douglas, André Henriques, Michael Hill, Homological obstructions to string orientations (arXiv:0810.2131)

• 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]