Twisted spin bordism invariants in 5 dimensions

Question

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance!

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The spin $G$-bordism invariant can be twisted in the way that the Spin($d$) group of the $d$-manifold and the $G$ group can be combined and mod out any shared normal subgroup.

For example, we can consider $Spin(d) \times_N G \equiv \frac{Spin(d) \times G }{N}$.

Here we denote $G_1 \times_N G_2 \equiv \frac{G_1 \times G_2 }{N}$ in general.

I am trying to understand a particular twisted spin bordism invariant in 5 dimensions, based on a computation of Adams spectral sequence.

I have obtained that 5d bordism groups as

$$ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{2}}= 0, $$ $$ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{4}} =\mathbb Z_{16}, $$ $$ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}, $$

My question: What are the above $\mathbb Z_{32}$ generator, $\mathbb Z_{16}$ generator, and $\mathbb Z_{2}$ generator as

(1) the 5d topological terms, and

(2) its 5d manifold generators?

What I have done which leads to a tentative clue for an answer:

• 1. I find Adams $\mathcal A$ module structure has the following for $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{4}} =\mathbb Z_{16}$ and $ \Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}$,
• 2. The $\mathbb Z_{16}$ in $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{4}} =\mathbb Z_{16}$ is similar to the $$ \Omega_4^{Pin^+}(pt) =\mathbb Z_{16}$$, which is well-known to be generated by a 4d $\eta$-invariant that can be obtained from a Dirac spinor in 4d. $$ \text{the $\eta$-invariant of the Dirac operator acting on the (twisted) Dirac spinor bundle} $$ Thus $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{4}} =\mathbb Z_{16} $ may be generated by $$A \cup \eta?$$ except that there is only an $A \in H^1(B \mathbb Z_2,\mathbb Z_2)=\mathbb Z_2$, there is no ${\mathbb Z_{16}}$ class of $A \cup \eta$?
• 3. The $\mathbb Z_{2}$ in $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}$ may be generated by $$ A' B' \cup \text{Arf} $$ where $A' \in H^1(B \mathbb Z_{4}, \mathbb Z_{2})=\mathbb Z_{2}$ and $B' \in H^2(B \mathbb Z_{4}, \mathbb Z_{2})=\mathbb Z_{2}$.
• 4. The $\mathbb Z_{32}$ in $\Omega_5^{Spin \times_{\mathbb Z_2} \mathbb Z_{8}}= \mathbb Z_{32} \times \mathbb Z_{2}$ may be related to the earlier $\mathbb Z_{16}$ and has something to do with the Postnikov square.

Answer 1

$\newcommand{\Z}{\mathbb Z}\newcommand{\RP}{\mathbb{RP}}$ Let $G_1 := \mathrm{Spin}_5\times_{\Z/2}\Z/4$. The group $\Omega_5^{G_1}$ is discussed by Tachikawa-Yonekura, §3.1 and §3.4. Given a 5-manifold $M$ with a $G_1$-structure given by the principal $G_1$-bundle $P\to M$, there's a principal $\Z/2$-bundle $Q := P\times_{G_1} \Z/2\to M$. If $N\subset M$ is a representative of the Poincaré dual to $w_1(Q)\in H^1(M;\Z/2)$, then $N$ acquires a pin^$+$-structure. Then:

• The map $\Omega_5^{G_1}\to\Omega_4^{\mathrm{Pin}^+}\cong\Z/16$ sending $M$ to the bordism class of $N$ is an isomorphism; in particular, one can realize the isomorphism $\Omega_4^{\mathrm{Pin}^+}\to\Z/16$ as an $\eta$-invariant, and putting this together gives a complete bordism invariant for $\Omega_5^{G_1}$.
• A manifold generator of $\Omega_5^{G_1}$ is $\RP^5$ with a $G_1$-structure such that $Q$ is the nontrivial principal $\Z/2$-bundle; then an embedded $\RP^4\subset\RP^5$ represents the Poincaré dual of $w_1(Q)$ and $\RP^4$ acquires a pin^$+$-structure representing $1$ or $-1$ in $\Omega_4^{\mathrm{Pin}^+}\cong\Z/16$.

Let $G_2 := \mathrm{Spin}_5\times_{\Z/2}\Z/8$. Then $\Omega_5^{G_2}$ is discussed by Hsieh, §2.2. In particular:

• Given an element $R$ of the representation ring of $\Z/8$, Hsieh shows how to define an exponentiated $\eta$-invariant $\eta_R\colon\Omega_5^{G_2}\to\mathrm U_1$, and proves that these are complete invariants, i.e. there is some combination of $\eta$-invariants for some of these representations realizing the maps $\Omega_5^{G_2}\to\Z/32$ and $\Omega_5^{G_2}\to\Z/2$ that you're interested in.
• Lens spaces, with various $G_2$-structures, generate $\Omega_5^{G_2}$.

Given $R$ as above and a lens space $L(m, \vec a)$ with $G_2$-structure, Hsieh provides a formula (equation (2.35) in the paper) for $\eta_R(L(m, \vec a))$; this should make it possible to explicitly determine which $\eta$-invariants and which lens spaces you're looking for.