# Is a translation in a compact Lie group homotopic to the identity?

The following exercise is from Guillemin and Pollack, Differential Topology.

Show that the Euler characteristic of the orthogonal group (or any compact Lie group for that matter) is zero. Hint: Consider the left multiplication by an element $A\neq I$ in $O(n)$.

As it is stated, the first clue I had was to apply this proposition:

Let $X$ be a smooth compact orientable manifold. If $X$ admits a smooth map $f:X\to X$ that is homotopic to the identity and has no fixed points then the Euler characteristic of $X$ is zero.

Of course, as the hint suggests we define $f:O(n)\to O(n)$ by $f(B)=AB$ for every $B\in O(n)$, and $A\in O(n)\setminus\{I\}$ is fixed. Clearly $f$ has no fixed points.

So, my question now is if $f$ is homotopic to the identity map? How can we show this?

Thank you very much.

• Well, I mean, this is not literally true. They need to demand the Lie group is positive-dimensional. In that case, pick an element in the path component of the identity to translate with. – user98602 May 13 '16 at 19:01
• Oh, I see. That sounds good. Thank you. But how do you know the path component of the identity is not a singleton? @MikeMiller – Talexius May 13 '16 at 19:03
• Use the positive-dimensional assumption. There's something to do here, so I'll let you think about it. – user98602 May 13 '16 at 19:13
• Oh, sorry. I don't know what I was thinking. Every Lie group is locally euclidean so of course path components are not trivial. @MikeMiller – Talexius May 13 '16 at 19:36
• You should write up an answer with all the details; I'll upvote it. – user98602 May 13 '16 at 19:36

It is suffice to show some non-trivial left-translation is homotopic to the identity map.

Let $G$ be a Lie group positive-dimensional. For every $x\in G$ we can find an open neighborhood of $x$ homeomorphic to an open ball of $\mathbb{R}^n$ ($\dim G= n>0$), which is path connected. Hence the path component of $x$ is always non-trivial. (Another way to see this is that as $G$ is locally path connected, its path components are open, then they are non-trivial.)

Let $e\in G$ be the identity element. From above, we can find $x\neq e$ and a continuous $\gamma:[0,1]\to G$ such that $\gamma (0)=e$ and $\gamma (1)=x$. Define the left-translation by $L_x(g)=xg$ for every $g\in G$. The function $H:G\times [0,1]\to G$ defined by $H(g,t)=\gamma(t)g$ is an homotopy between $L_x$ and the identity map.