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Suppose we consider a $d$-cobordism group classifying manifolds with $H$-structure and with a classifying space $BG$ of a group $G$, written as $$ \Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots, $$ where there is a particular $\mathbb{Z}_m$-class (a torsion part $\mathbb{Z}_m$) we focus on.

For simplicity, for your convenience, we can, for example, focus on $d=3$ case and $H=Spin$. Namely, we can focus on spin-cobordism.

Let us try to relate a cobordism generator (say some topological invariants) in this particular $\mathbb{Z}_m$-class, say a $\nu \in \mathbb{Z}_m$-class, to one higher dimensional cobordism group. Let $\nu$ be any divisor of $m$, thus $\nu \mid m$ is true, thus $\gcd(\nu , m)=\nu $.

I wonder whether

  1. If we consider this $(d+1)$-cobordism group $$ \Omega^{d+1}_{Spin}(B(\mathbb{Z} \times G))=\Omega^{d+1}_{Spin}(B\mathbb{Z} \times BG))= \Omega^{d+1}_{Spin}(S^1 \times BG)) \overset{?}{=} \mathbb{Z}_{\frac{m}{\nu}} \oplus \dots? $$ Do we necesssarily obtain a $\mathbb{Z}_{\frac{m}{\nu}}$-subclass deriavable out of $\mathbb{Z}_m$-class from $\Omega^{d}_{Spin}(BG)= \mathbb{Z}_m$? (Recall $\nu \mid m$, so $\frac{m}{\nu}$ is a natural number.) More generally, it can be that $\mathbb{Z}_{n} \supseteq \mathbb{Z}_{\frac{m}{\nu}}$ where $\mathbb{Z}_{\frac{m}{\nu}}$ is a normal subgroup of $\mathbb{Z}_{n}$, $$ \Omega^{d+1}_{Spin}(B(\mathbb{Z} \times G))=\Omega^{d+1}_{Spin}(B\mathbb{Z} \times BG))= \Omega^{d+1}_{Spin}(S^1 \times BG)) \overset{?}{=} \mathbb{Z}_{n} \oplus \dots? $$ Are there certain simple criteria that the relation holds to be true? And there certain simple criteria that the relation holds to be false?

Note that we had simplified $B\mathbb{Z}=S^1$.

  1. If we replace the above $B\mathbb{Z}$ to $B\mathbb{Z}_{\frac{m}{\nu}}$, do we still have a similar result? $$ \Omega^{d+1}_{Spin}(B(\mathbb{Z}_{\frac{m}{\nu}} \times G))\overset{?}{=} \mathbb{Z}_{\frac{m}{\nu}} \oplus \dots ? $$

In some sense, my conjecture above may have something to do with a simple version of categorification?

Any existent literature/References are welcome! Thank you for the comments and References!

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  • $\begingroup$ I don't understand what the connection to categorification is supposed to be? $\endgroup$ – Tim Campion Dec 27 '18 at 16:24

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