Definition of the higher dimensional mapping tori

Question

This is proving harder to search for than I imagined. The usual definition of a mapping torus $\mathcal{M}_h$ associated to a homeomorphism $h\colon X\rightarrow X$ on a topological space $X$ is the quotient of the product $X\times I$ by the equivalence relation given by $(x,1)\sim_h (h(x),0)$. So $$\mathcal{M_h}=(X\times I)/\sim_h$$ As $h$ is invertible and can be iterated, this space acts as a model to study the dynamics of the $\mathbb{Z}$ action on $X$ given by $h$.

I've recently found it necessary to study a higher analogue of this construction defined by a $G=\mathbb{Z}^d$ action of homeomorphisms on a space $X$. More precisely, I would like the higher dimensional mapping torus $\mathcal{M}_G$ to fiber over the $d$-dimensional torus with fiber $X$ and such that an $\mathbb{R}^d$ action on $\mathcal{M}_G$ is inherited such that the subaction by the subgroup $\mathbb{Z}^d$ fixes fibers set-wise and acts as $G$ on individual fibers. That is, $\mathcal{M}_G$ should model the dynamics of the action of $G$ on $X$.

However, the exact formulation of this construction is not entirely obvious to me. My attempt at giving a definition is as follows.

Let $G\cong \mathbb{Z}^d$ act on the space $X$ by homeomorphisms and let $\mathbb{Z}^d$ be generated by the $d$ homeomorphisms $h_i\colon X\rightarrow X$, with $1\leq i\leq d$. We form the mapping torus $\mathcal{M}_G$ associated to the action of $G$ on $X$ by taking the quotient of the product $X \times [0,1]^d$ by the equivalence relation generated by $\sim_G$ where, for all $x\in X$, $(x, s_1, \ldots, s_d) \sim_G (h_i(x), t_1, \ldots, t_d)$ if for some $1\leq i\leq d$ $s_i=1$, $t_i=0$ and $s_j=t_j$ for all $j\neq i$. That is, $$\mathcal{M}_G=(X\times [0,1]^d)/\sim_G.$$

Is this the usual construction, or is there a simpler formulation? Or does the above not even coincide with the usual definition? (I'm thinking that maybe there is a nicer way of forming this space as a covering by the space $X\times\mathbb{R}^d$). I would also very much appreciate if someone could suggest places in the literature where this construction is described and used.

Answer 1

A colleague pointed out the following article$^{[1]}$ which constructs essentially the same object as was described in the OP.

Suppose that $T_1,\ldots T_d$ are commuting homeomorphisms of $X$ which generate a $\mathbb{Z}^d$-action, $\alpha\colon\mathbb{Z}^d\times X\rightarrow X$.

Definition $2.4$ Given a $\mathbb{Z}^d$-action $\alpha$ on $X$, we write $M_d(X)$ for the mapping torus $\alpha$. The twisted product space $\mathbb{R}^d\times_{\alpha} X$ is given by factoring the product space $\mathbb{R}^d\times X$ by the relations $$(b_1,\ldots,b_{i-1},b_i,b_{i+1},\ldots,b_d;x)\sim (b_1,\ldots,b_{i-1},b_i-1,b_{i+1},\ldots,b_d;T_i(x))$$

This gives the same space as was described in the OP.

$[1]$ ALAN FORREST and JOHN HUNTON (1999). The cohomology and $K$-theory of commuting homeomorphisms of the Cantor set. Ergodic Theory and Dynamical Systems, 19, pp 611-625.