Linear action induced in homology of the $n$-dimensional torus

I'm reading a lecture notes on actions of $$\mathbb{Z}^{p}$$ on the torus but I think that are not selfcontained. I need to know the definition of the "induced action in homology" of a given action.

Let $$p,q\in\mathbb{N}$$ and $$T=\mathbb{R}^{q}/\mathbb{Z}^{q}$$ the $$q-$$dimensional torus. A linear action of the aditive group $$\mathbb{Z}^{p}$$ is any homomorphism of groups $$\phi:\mathbb{Z}^{p}\rightarrow End(T^{q})$$. Then, $$\phi$$ induces a linear action $$\phi_{*}$$ of $$\mathbb{Z}^{p}$$ on the first homology group $$H_{1}(T^{q},\mathbb{Z})$$. My question is

Which is the definition of $$\phi_{*}$$?

I will appreciate any reference on this particular topic.

• any map of spaces induces a map on homology Dec 27, 2021 at 21:49
• Do you have a link (or title & author) for the mentioned lecture notes? Dec 28, 2021 at 4:41

Roughly speaking, any element $$\gamma$$ of $$H_1(\mathbb{T}^q;\mathbb{Z})$$ can be thought of as a continuous loop in $$\mathbb{T}^q$$ (more accurately elements of $$H_1$$ are homotopy classes of such loops). In particular any such $$\gamma$$ can be written in the form

$$\gamma=\sum_{i=1}^q z_i \gamma_i,$$

where $$z_i$$'s are integers and for $$1\leq i\leq q$$, $$\gamma_i: [0,1]\to \mathbb{T}^q$$ is defined by $$\gamma_i(t)=(0,\cdots,0,t,0,\cdots,0)$$, where $$t$$ is in the $$i$$-th coordinate.

(See e.g. What does the element of homology groups mean? for a more accurate description.)

Example 1: For $$q=2$$, thinking of $$\mathbb{T}^2$$ as a square with opposite sides identified, $$\gamma_1$$ is the horizontal loop, $$\gamma_2$$ is the vertical loop, $$\gamma_1+\gamma_2$$ is the diagonal loop from the bottom left corner to top right corner and $$-\gamma_1+\gamma_2$$ is the diagonal loop from the bottom right corner to the top left corner.

After fixing the coordinate loops $$\gamma_1,\cdots, \gamma_q$$ since the integers $$z_1,...,z_q$$ completely determine $$\gamma$$ we have an isomorphism of groups:

$$H_1(\mathbb{T}^q;\mathbb{Z})\stackrel{\cong}{\to} \mathbb{Z}^q,\,\, \sum_{i=1}^q z_i \gamma_i \mapsto (z_1,\cdots ,z_q).$$

Next, if $$f:\mathbb{T}^q\to \mathbb{T}^q$$ is a continuous map (it does not need to be an endomorphism) and if $$\gamma$$ is a continuous loop in $$\mathbb{T}^q$$, then $$f\circ \gamma$$ is also a continuous loop in $$\mathbb{T}^q$$. Thus (as $$f$$ also preserves homotopies) there is an induced map $$f_\ast: H_1(\mathbb{T}^q;\mathbb{Z})\to H_1(\mathbb{T}^q;\mathbb{Z})$$.

Example 2: In a recent answer (https://math.stackexchange.com/a/4320757/169085) I've established that any Lie group endomorphism of $$\mathbb{T}^q$$ is given by a $$q\times q$$ matrix with integer entries. Fix $$q=2$$ and consider the endomorphism $$f:\mathbb{T}^2\to\mathbb{T}^2, (x,y)\mapsto (2x,-3y)$$. To see what the induced map $$f_\ast$$ does on homology, consider the coordinate loops:

$$f_\ast(\gamma_1)(t)=f\circ \gamma_1(t)=(2t,0)=2\gamma_1(t);\,\, f_\ast(\gamma_2)(t)=f\circ \gamma_2(t)=(0,-3t)=-3\gamma_2(t).$$

Thus $$f_\ast$$ takes the horizontal coordinate loop $$\gamma_1$$ and streches it by a factor of two so that $$f_\ast(\gamma_1)$$ is still horizontal but wraps around twice. Similarly $$f_\ast$$ takes the vertical coordinate loop $$\gamma_2$$, stretches it by a factor of three so that $$f_\ast(\gamma_2)$$ is still vertical, and also reverses its orientation. In terms of the isomorphism $$H_1(\mathbb{T}^2;\mathbb{Z})\cong \mathbb{Z}^2$$, $$f_\ast$$ is exactly the matrix $$\begin{pmatrix}2&0\\0&-3\end{pmatrix}$$ by which the original toral endomorphism $$f$$ is given.

(It might be a good idea to relate the four loops in Example 1 to each other by way of toral endomorphisms; e.g. what is an example of a toral endomorphism $$g$$ with $$g_\ast(\gamma_1)=-\gamma_1+\gamma_2$$ etc.)

Finally note that if $$f$$ and $$g$$ are both continuous self-maps of $$\mathbb{T}^q$$, then so is $$f\circ g$$, and $$(f\circ g)_\ast(\gamma)=(f\circ g)\circ \gamma=f\circ (g\circ \gamma)=f_\ast(g_\ast(\gamma))=(f_\ast\circ g_\ast)(\gamma)$$, so that the process of "inducing on homology" preserves compositions of functions. Thus any group action $$\phi: \mathbb{Z}^p\to\operatorname{Homeo}(\mathbb{T}^q)$$ by homeomorphisms (and in particular any group action $$\phi: \mathbb{Z}^p\to\operatorname{Aut}_{\text{Lie}}(\mathbb{T}^q)$$ by Lie group automorphisms) induces a group action $$\phi_\ast: \mathbb{Z}^p \to \operatorname{Aut}_{\text{Ab}}(H_1(\mathbb{T}^q;\mathbb{Z}))$$ by group automorphisms.

• Thank you so much. This is a very clear answer. Dec 28, 2021 at 17:20
• @JoséLuisCamarilloNava I'm glad it was useful. Like I asked above I would be interested in having a look at the lecture notes you are reading, if they are publicly available. Dec 28, 2021 at 17:32