- It is said that the Pontrjagin dual of the 3-dimensional Spin bordism group of $BG$ for $G$ a finite group,
$$
\text{Hom}(Ω^{spin}_3(BG),\mathbb{R/Z}),
$$
can be expressed by triples of cochains $$(w, p, a),$$
where 3-cochain $$w \in C^3(BG; \mathbb{R/Z}),$$
2-cocycle
$$p \in Z^2(BG;\mathbb{Z}/2)$$ and $a$ is a 1-cocycle
$$a \in Z^1(BG;\mathbb{Z}/2),$$
satisfying the equation
$$dw + (1/2) p + (1/4) P(a^2) = 0.$$
Here $P$ is the Pontrjagin square. Furthermore, by redefinition, we can use a different triple, $$(w', p, a)=(w-A^3/8,p,a),$$ satisfying the equation
$$dw' + (1/2) p^2 = 0,$$
where 1-cochain $A \in C^1(BG;\mathbb{Z}/2)$ is the integral lift of $a$.
- It is said that the Pontrjagin dual of the 4-dimensional Spin bordism group of $BG$ for $G$ a finite group,
$$
\text{Hom}(Ω^{spin}_4(BG),\mathbb{R/Z}),
$$
can be expressed by triples of cochains $$(u, q, b),$$
where 4-cochain $$u \in C^4(BG; \mathbb{R/Z}),$$
3-cochain
$$q \in C^3(BG;\mathbb{Z}/2)$$ and $b$ is a 2-cocycle
$$b \in Z^2(BG;\mathbb{Z}/2),$$
satisfying the equation
$$du + b^2 = 0,$$
perhaps with more constraints (to be confirmed?).
Question:
- Is there some more concise and simple constraint for the triple of
$$ \text{Hom}(Ω^{spin}_4(BG),\mathbb{R/Z})? $$- Is there an easy way to related the triple data of $ \text{Hom}(Ω^{spin}_3(BG),\mathbb{R/Z}) $ to the triple data of $ \text{Hom}(Ω^{spin}_4(BG),\mathbb{R/Z}) $? For example, by a dimensional reduction from 4d to 3d?
Refs: See for example,
