# Relating bordism groups of different dimensions

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.

• What do you mean by $M_{d-n}$ generator of bordism group of Eilenberg-Maclane space? Oct 24, 2018 at 18:36

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.