# Fivebrane bordism $\Omega_d^{\mathrm{Fivebrane}}$

$$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$$What are the (lower-dimensional, say $$d \leq 14$$) Fivebrane bordism groups $$\Omega_d^{\Fivebrane}=\text?$$

Given that we know the Whitehead tower

$$\require{AMScd}\begin{CD} \vdots \\ @VVV \\ B\Fivebrane \\ @VVV \\ B\String @>\frac1 6 p_2>> B^7U(1) @>\cong>> B^8\mathbb Z \\ @VVV \\ B{\Spin} @>\frac1 2 p_1>> B^3U(1) @>\cong>> B^4\mathbb Z \\ @VVV \\ B{\SO} @>w_2>> B^2\mathbb Z_2 \\ @VVV \\ BO @>w_1>> B\mathbb Z_2 \\ @VV\cong V \\ B{\GL} \end{CD}$$

and the string bordism groups:

\begin{align*} & \Omega_0^{\String}=Z \\ & \Omega_1^{\String}=Z_2 \\ & \Omega_2^{\String}=Z_2 \\ & \Omega_3^{\String}=Z_{24} \\ & \Omega_4^{\String}=0 \\ & \Omega_5^{\String}=0 \\ & \Omega_6^{\String}=Z_2 \\ & \Omega_7^{\String}=0 \\ \end{align*}

Framed bordism: $$\Omega_7^{\Fr}=Z_{240}.$$

Note that $$\Omega_d^{\String}=\Omega_d^{\Fr}, d \leq 6.$$

Is it true that $$\Omega_d^{\Fivebrane}=\Omega_d^{\Fr}, d \leq 7?$$ How about $$\Omega_d^{\Fivebrane}=\text?, \text{ for } d > 7?$$

This should be a comment but turns out too long...

According to Bott's periodicity, the next layer of your Whitehead tower is $$\cdots\to \mathrm{MFivebrane}\to B^9\mathbb{Z}/2$$. Hence your guess is true. Moreover, $$\Omega^{\mathrm{fr}}_{8}\to\Omega^{\mathrm{Fivebrane}}_{8}$$ is an epimorphism.

The reason is as follows. Consider a stable tangential structure $$S$$ in your Whitehead tower. Recall that the Thom spectrum $$\mathbb{M}S$$ is defined by the Thom spaces of the canonical vector bundle on $$BS(n)$$ and the natural embeddings. Therefore, the connectedness of the natural spectrum map $$\mathbb{S}\to\mathbb{M}S$$ is determined by the connectedness of $$BS$$. More precisely, $$\mathbb{S}\to\mathbb{M}S$$ is $$(n\!-\!1)$$-connected if $$BS$$ is $$(n\!+\!1)$$-connected. This implies that $$\Omega^{\mathrm{Fr}}_{\bullet}\equiv\mathbb{S}_{\bullet}\to\Omega^{S}_{\bullet}\equiv\mathbb{M}S_{\bullet}$$ is an isomorphism for $$\bullet\leq n$$ and is an epimorphism for $$\bullet=n\!+\!1$$.

As for the general structure of $$\Omega^{\mathrm{Fivebrane}}_{\bullet}$$, I'm not aware of a computation in the literature. I'm also interested in knowing about it.

• May I ask if there's a difference between $\Omega^{\mathrm{Fr}}$ and $\Omega^{\mathrm{fr}}$ (some kind of connectivity?) or was that inadvertent? Commented May 20 at 5:23
• @DavidRoberts They are the same. Sorry for being careless.
– Leo
Commented May 20 at 9:33
• not a problem! I edited to make them the same, but then I know there's a big difference between eg Tmf and tmf, so I made sure I didn't submit that change. Commented May 20 at 12:27