# Prove that the map $c:{\mathbb {T}}^{3}=S^{1}\times S^{1}\times S^{1}\setminus\ \Delta\longrightarrow \{\pm 1\}$ is continuous.

Prove that the map $$c:{\mathbb {T}}^{3}=S^{1}\times S^{1}\times S^{1}\setminus \Delta\longrightarrow \{\pm 1\}$$ is continuous where $$\Delta$$ is the diagonal $$\Delta=\{(g_i,g_j,g_k)\}$$ for $$i=j$$ or $$j=k$$ or $$i=k$$ and $$c$$ has the following properties:

1)($$\textit{Left-Invarianc}$$): $$c(ag_1,ag_2,ag_3)=c(g_1,g_2,g_3)$$ for all $$a,g_1,g_2,g_3\in S^1$$ with $$g_i\ne g_j$$ for any $$i$$ and $$j$$

2)($$\textit{Co-cycle condition}$$): $$c(g_1,g_2,g_3)-c(g_1,g_2,g_4)+c(g_1,g_3,g_4)-c(g_2,g_3,g_4)=0$$ for all $$g_1,g_2,g_3,g_4\in S^1$$ with $$g_i\ne g_j$$ for any $$i$$ and $$j$$

These two conditions basically tells us that there is a certain order on circle which is left-invariant.

I want to prove or disprove the continuity of $$c$$ where $$\{\pm 1\}$$ is given discrete topology. We see that $$\mathbb{T}^{3}=S^{1}\times S^{1}\times S^{1}$$ is a three dimensional torus and not much of its topological facts are given in the wiki article. Please suggest where should I start.

EDIT: Image after Paul Plummer's comment

• Try to check what the connected components of $\mathbb T^3 \setminus \Delta$ are.
– Levi
Nov 2, 2019 at 21:04
• @Levi How exactly? This three torus blows my mind when I try to visualize.
– user716941
Nov 3, 2019 at 0:14
• When you say that "prove that the map $c$..." Did you mean "prove that any map $c$... With the following properties..." I originally read it as wanting to show the standard circular order is continuous but it seems, from what you said in the comments, that is not what you meant.
– user29123
Nov 3, 2019 at 17:36

You can choose such a $$c$$ to be continuous. Note that means that the 3-torus without the fat diagonal is at least two connected components. How can we see that?

First, instead of trying to imagine a 3-torus, imagine three, possibly with multiplicity, marked points (which are colored/distinguished from each other) on the circle $$S^1$$ which will correspond to a point in $$\mathbb T^3$$. So if you "visually" see only two or one marked point that means that some marked points are overlapping (two coordinates are equal) and so is a point in the fat diagonal $$\Delta$$. Lets call the marked points $$r,b,g$$ and placed at $$-1,i,1$$ in $$S^1$$. It should be clear that you can not move the marked points so that $$r,b,g$$ are $$-1,-i,1$$ without crossing the fat diagonal.

It should also be clear, with this interpretation, that the components are fixed by the $$S^1$$ action, so $$c$$ can assigns the connected components a single value hence the function is continuous.

• This is a place to start (as asked for) and how to think about it. My answer is of course not a full proof but it has the ideas of how to approach it.
– user29123
Nov 2, 2019 at 21:37
• I do not understand your argument quite well. I didn't understand this line, "So if you "visually" see two or one marked point that means that is a point in the fat diagonal Δ." and from your last paragraph, "with this interpretation, that the components are fixed by the $S^1$ action". Can you please explain it further. I have added the diagram to the original post after what I have understood from your suggestion.
– user716941
Nov 3, 2019 at 0:12
• @mathwizard If you move the marked points around so that two overlap that corresponds to two coordinates being equal, so you would be in the fat diagonal (not seeing a third point on the circle). The $S^1$ action is just rotation of the circle, and choosing three points on the circle and rotating gives a path from one point in the component to another point.
– user29123
Nov 3, 2019 at 0:22
• Also your picture isn't quite right, a single point in the torus corresponds to three points (with multiplicity) on the circle
– user29123
Nov 3, 2019 at 0:23
• Yes picture is not right. Thanks. I am wondering how do we utilize those two conditions (1) and (2) of function $c$ to prove its continuity in the sense you have interpreted it ?
– user716941
Nov 3, 2019 at 3:23