Relating bordism groups of different dimensions

Question

Let

$M_d$

be a $d$-manifold generator of a subgroup of bordism group $$ \Omega_d^{G}, $$ or further generalization

$$ \Omega_d^{G}(K(\mathcal{G},n+1)), $$

which $G$ is the given structure including the tangent bundle structure (such as the SO or Spin) and the internal gauge bundle structure (such as an additional compact Lie group). The $K(\mathcal{G},n)$ is the Eilenberg–MacLane space of $\mathcal{G}$; if $n$ > 1, then $\mathcal{G}$ must be abelian.

Generally, $M_d$ cannot be written as $M'_{d-1} \times S^1$, nor $\tilde M_{d-n} \times T^n$, where $T^n$ is the $n$-torus.

• Here are my questions:

(1) However, are there certain cases that $$M_d \overset{?}{=}M_{d-n} \times T^n,$$ such that $M_{d-n}$ is also a $(d-n)$-manifold generator of a subgroup of bordism group $$ \Omega_{d-n}^{G}(K(\mathcal{G},1))? $$ Now $K(\mathcal{G},1)=B\mathcal{G}$ is the classifying space of $\mathcal{G}$.

(2) Are there actually mathematical proofs or theorems stating the similar structures given above relation the bordism group generators of $\Omega_d^{G}(K(\mathcal{G},n+1))$ to the bordism group generators of $\Omega_{d-n}^{G}(K(\mathcal{G},1))$, for whatever integer $n$? If so, please state the results and please provide the Refs?

Many thanks.

Answer 1

Just a few remarks:

1. Bordism of Eilenberg--Mac Lane spaces can be identified with bordism of pairs consisting of a closed manifold with a cohomology class. More precisely, $\Omega^G_d(K(\mathcal{G},n+1))$ can be described as bordism classes of pairs $(M^d,x)$ where $M^d$ is a closed, $d$-dimensional $G$-manifold and $x\in H^{n+1}(M^d;\mathcal{G})$ is a cohomology class.
2. For any (reduced) generalized homology theory $E_*$ there is a loop-suspension homomorphism $$ E_k(\Omega X)\to E_{k+1}(\Sigma\Omega X)\to E_{k+1}(X). $$ In particular, since $K(\mathcal{G},i)=\Omega K(\mathcal{G},i+1)$ for $i\ge1$ there are homomorphisms $$ \Omega^G_{d-n}(K(\mathcal{G},1))\to \Omega^G_{d-n+1}(K(\mathcal{G},2))\to \cdots \to \Omega^G_{d}(K(\mathcal{G},n+1)). $$
3. One might expect that the homomorphisms $$\Omega^G_{d-n+i-1}(K(\mathcal{G},i))\to \Omega^G_{d-n+i}(K(\mathcal{G},i+1))$$ appearing in 2. are described geometrically by taking the bordism class of a pair $(M,x)$ to that of the pair $(M\times S^1,x\times \sigma)$, where $\sigma\in H^1(S^1;\mathbb{Z})$ denotes a generator and $\times$ denotes cross product. Something like this must be true, but care needs to be taken with reduced vs. unreduced homology theories.