Csdrhrt

question 1: Are there mathematical definition of the semi-direct product between manifolds
$$
M^{d_1} rtimes V^{d_2}?
$$

For example, is it defined as a fibration such that $M^{d_1}$ is the fiber and the $V^{d_2}$ is the base, so the total space is a bundle with the following relation
$$
M^{d_1} hookrightarrow M^{d_1} rtimes V^{d_2} to V^{d_2}
$$

question 2: Can $M^{d_1} rtimes V^{d_2}$ be a mapping torus?

question 3: What is the mapping class group of $M^{d_1} rtimes V^{d_2}$?

Partial answers are very welcome! Thanks!

I don't know any standard way to define this semidirect product of manifolds, and I don't see how we can invent one.

Your intuition is correct that a fiber bundle of manifolds is like a product manifold but with more complex structure, roughly in the same way that a semidirect product of groups is like a product group but with more complex structure.

However, I think the analogy breaks down. In the case of groups, when we have $G = N rtimes_phi H$, we are also given an action $phi$ of $H$ on $N$. In the case of manifolds $M$ and $V$, in order for $V$ to act on $M$, $V$ needs some algebraic structure. We could require $V$ to be a Lie group (basically, a group which is also a manifold), but I still don't see how to define the total space.

A standard mapping torus on $M$ is a fiber bundle with fiber $M$, but its base space is a circle whereas you want it to be a more general $V$.