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.