isotopy of homeomorphisms of a torus

Question

Let's consider some homeomorphism of a torus which is isotopic to identity. Is it possible to construct an explicit isotopy?

Edit: It's well-known statement that a homoemorphism of a torus is isotopic to the identity if and only if it induces the trivial automorphism of a fundamental group. Could you please show me the proof?

Answer 1

Here is the answer for all tori $T^n$ except in dimension 4:

1. If $n\le 3$ then a homeomorpism $f: T^n\to T^n$ is homotopic to the identity (equivalently, $f$ induces the identity automorphism of $\pi_1(T^n)$) if and only if $f$ is isotopic to the identity. For $n=2$ you can find a proof for instance in

B. Farb and D. Margalit, "Primer on mapping class groups". (The same "homotopy implies isotopy" holds for all closed surfaces).

For $n=3$, this is a special case of a theorem by Waldhasen about homeomorphisms of a much larger class of "Haken" 3-manifolds,

F. Waldhausen, "On irreducible 3-manifolds which are sufficiently large", Ann. of Math. (2) 87 (1968) 56–88.

2. For all $n\ge 5$, there exists a natural short exact sequence $$ 1\to Z_2^{\infty} \to Map(T^n) \stackrel{\phi}{\to} GL(n,Z)\to 1, $$ where $$ Map(T^n)=Homeo(T^n)/\sim$$ and $\sim$ denotes isotopy, and the homomorphism $\phi$ is defined by the action of homeomorphisms $T^n\to T^n$ on the fundamental group of the torus.

A proof can be found in Theorem 4.1 of

A. Hatcher, "Concordance spaces, higher simple-homotopy theory, and applications. Algebraic and geometric topology", Part 1, pp. 3–21, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.

In other words, for all $n\ge 5$, there exist infinitely many isotopy classes of homeomorphisms $T^n\to T^n$ which are all homotopic to the identity (homotopy does not imply isotopy).

Answer 2

It is easy to construct examples of isotopies on the $n$-torus $T^n = (S^1)^n$; all we have to do is work in the natural coordinate system induced by the identification of $S^1$ with the unimodular complex numbers. Indeed, representing points of $S^1$ as $e^{i\theta}$ where $0 \le \theta < 2\pi$, we see that for any real $\alpha$, the map $\phi_\alpha: S^1 \to S^1$ given by $\phi_\alpha(e^{i\theta}) = e^{i(\theta + \alpha)}$ is surjective, injective, differentiable, and has a differentiable, bijective inverse which is fact given by $\phi_{-\alpha}(e^{i\theta}) = e^{i(\theta - \alpha)}$. All these assertions follow readily from the elementary properties of multiplication in the complex field; of course it must be remembered that we need identify values of $\theta + \alpha$ modulo $2\pi$ in accord with the fact that $e^{2\pi i} = 1$. It is easy to see that each $\phi_\alpha$ is homotopic to the identity map through a family of diffeomorphisms; indeed we may take $\Phi_{\alpha, t}(e^{i\theta}) = e^{i(\theta + t\alpha)}$ where $t \in [0, 1]$; then $\Phi(\alpha, 0)(e^{i\theta}) = e^{i\theta}$ is the identity map and $\Phi(\alpha, 1) = e^{i(\theta + \alpha)}= \phi_\alpha(e^{i\theta})$; clearly $\Phi(\alpha, t)$ is a diffeomorphism for every $t \in [0, 1]$. Thus $\Phi(\alpha, t)$ provides an example of an isotopy of the one-torus $S^1$.

The preceeding example may easily be generalized to the case of the $n$-torus $T^n = (S^1)^n$ by simply using a separate complex coordinate $e^{i\theta_j}$ for each of the $n$ copies of $S^1$ in the product $T^n$. Then we may, for a vector $\bar \alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n)$, define a diffeomorphism $\phi_{\bar \alpha}:T^n \to T^n$ by $\phi_{\bar \alpha}(e^{i\theta_1}, e^{i\theta_2}, \ldots, e^{i\theta_n}) = (e^{i(\theta_1 + \alpha_1)}, e^{i(\theta_2 + \alpha_2)}, \ldots, e^{i(\theta_n + \alpha_n)})$ and an isotopy $\Phi_{\bar \alpha, t}(e^{i\theta_1}, e^{i\theta_2}, \ldots, e^{i\theta_n}) = (e^{i(\theta_1 + t\alpha_1)}, e^{i(\theta_2 + t\alpha_2)}, \ldots, e^{i(\theta_n + t\alpha_n)})$. Since each coordinate map $e^{i\theta_j} \to e^{i(\theta_j + t \alpha_j)}$ is independent of the others, the injectivity, surjectivity, differentiability, and differentiable invertability follow exactly as in the case of $S^1$. We have thus shown, by explicit construction, the existence of diffeomorphisms of $n$-tori which are isotopic to the identity map.

In fact the maps $\Phi_{\bar \alpha, t}$ are an extremely specialized instance of much more general classes of isotopic diffeomorphisms. Perhaps one of the best ways to see this is to recall that, for compact differentiable manifolds $M$ and vector fields $X$ on $M$, the flow $\psi_t$ of $X$ is a one-parameter family of diffeomorphisms of $M$ such that $d\psi_t(x)/dt = X(\psi_t(x))$ and $\psi_0(x) = x$ for all $x \in M$. As is well-known, the flow $\psi_t$ represents the complete set of integral curves of the vector field $X$; for each $t \in \Bbb R$, $\psi_t:M \to M$ is a diffeomorphism. Since $\psi_0$ is the identity map on $M$, we see that each $\psi_t$ is isotopic to the identity map. In the case at hand, we may indeed compute

$\dfrac{d\Phi_{\bar \alpha, t}(e^{i\theta_1}, e^{i\theta_2}, \ldots, e^{i\theta_n})}{dt} = \dfrac{d(e^{i(\theta_1 + t\alpha_1)}, e^{i(\theta_2 + t\alpha_2)}, \ldots, e^{i(\theta_n + t\alpha_n)})}{dt}$ $= (i\alpha_1 e^{i(\theta_1 + t\alpha_1)}, i \alpha_2 e^{i(\theta_2 + t\alpha_2)}, \ldots, i\alpha_n e^{i(\theta_n + t\alpha_n)}); \tag{1}$

setting $t = 0$ shows that $\Phi_{\bar \alpha, t}$ is in fact the flow of the vector field whose value at $(e^{i\theta_1}, e^{i\theta_2}, \ldots, e^{i\theta_n})$ is $(i \alpha_1 e^{i\theta_1}, i \alpha_2 e^{i \theta_2}, \ldots, i \alpha_n e^{i \theta_n})$. This may look more familiar if expressed in real coordinates via the use of Euler's identity $e^{i \theta} = \cos \theta + i \sin \theta$; I leave such a transformation to those of my readership interested in such details.

Hope this helps. Cheerio,

and as ever,

Fiat Lux!!!