Categorification-like statement in the cobordism group?

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!

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