$\pi_1 $ of space of positive-definite matrices

Question

I am trying to find the fundamental group of the space of positive-definite matrices that are also symmetric with entries that are real. Call this space $M$.

Where to begin? From particular examples, I think that this group is the identity group $[e]$, but this is only intuition.
What do loops in in $M$ look like? I have an idea of what a path looks like.

Example: for $$\lambda \in [0,1]$$ we get $$x^T(\lambda A+(1-\lambda)B)x > 0$$ with $A$ and $B$ inside M and $x$ a vector, so $f(\lambda)=\lambda A+(1-\lambda)B$ is a path in $M$.

Will I need to consider van Kampen's theorems or is there a more direct way? Cheers

Answer 1

Since $M$ is convex, it is contractible and so its fundamental group is trivial. Explicitly, fix some $A\in M$ and define $f:M\times[0,1]\to M$ by $f(B,\lambda)=\lambda A+(1-\lambda)B$, and then $f$ is a contraction of $M$.