Motivated by the concept of diagonally dominated matrices we consider the space $S$ of all complex $n\times n$ matrices with $|a_{ii}|>\sum_{j\neq i} |a_{ij}|$, for every $i$. Every element of $S$ is invertible
Is $S$ a connected subset of $GL_{n}(\mathbb{C})$?What is its fundamental group?
One can consider the same question for real matrices with positive determinant(to avoid disconnecte-ness)
We can construct a homotopy retract of this to the set of diagonal matrices with nonzero diagonal entries by scaling nondiagonal entries to 0, and from there to a torus by scaling the diagonal entries to norm 1. Therefore, it is homotopy equivalent to $(S^1)^n$, and is connected.
For real matrices, the retract above still works, and takes you to $\{-1,1\}^n$. The subset with positive determinant is the subset with product 1, and is not connected for $n>2$. I think the connected component you may want is the subset where all eigenvalues have positive real part, but have not checked.
