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

Question

$\newcommand{\Fr}{\mathrm{Fr}}\newcommand{\Fivebrane}{\mathrm{Fivebrane}}\newcommand{\String}{\mathrm{String}}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\GL{GL}$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 $$ \Omega_d^{\Fivebrane}=\text?, \text{ for } d > 7? $$

Answer 1

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 $\bullet=n\!+\!1$.

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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.