Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant

Question

The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame subset of $\mathbb{R}^{n^2}$ i.e., if for every $\epsilon>0$ there exists a $\delta>0$ such that if the Euclidean norm $|x-y|<\delta$ for $x,y\in C$ then there is a path of length $<\epsilon$ in $C$ joining $x$ and $y$? An earlier version of the question used the term "locally path-connected as a subset" which some users found confusing. Note that local path-connectedness of subsets of Euclidean space is a completely standard usage. In particular the "comb" in the plane is not locally path connected in the plane precisely in this sense; see wiki.

Answer 1

The proposed fact is true, this is how I'd prove it.

The set $M_n(\mathbb R)\setminus\{0\}$ retracts on the unit sphere $S$ by linear rescaling $M\mapsto \frac{M}{||M||}$. Let $\hat C$ be the image of $C$.

Now $\Delta:=\{\det M=0\}$ is an algebraic hypersurface in $S$ (because the determinant is homogeneous) and $\hat C$ is a connected component of $S\setminus\Delta$. Let $d$ be the Euclidean distance induced on $S$.

Assume that $C$ is not tame. This gives a sequence $(M_n,N_n)_n$ of matrices in $S$ such that $d(M_n,N_n)<\frac{1}{n}$ and $M_n$ and $N_n$ cannot be joined by a path of length less than some $\epsilon$. It is so because what happens along half-lines $\mathbb R_{>0} M$ is tame. Up to extract a subsequence we can form the limit $M\in S$.

• Clearly $M\in\Delta$ for otherwise one can take a ball of small radius $r>0$ in $\hat C$ around $M$, and all points in this ball are linked by a path of length at most $Kr$ for some universal constant $K=K_n$.
• If $M$ belongs to a regular point of $\Delta$ then locally you can find an analytic rectification of $\Delta$ onto a coordinate axis in $\mathbb R^{n^2-1}$. Here again a small half-ball will give a contradiction (the fact that the rectification is sufficiently smooth implies that the distance is not too distorted). The point of the argument here is that you cannot "reach the other side" (even by taking a long detour) without crossing $\Delta$ itself.
• If $M$ belongs to the singular set of $\Delta$, you apply the same argument after desingularization of the hypersurface. According to this post the projection on the second factor $$R~:~\mathbb P_{n-1}(\mathbb R)\times S\longrightarrow S $$ is a resolution of singularities when restricted to the set $X:=\{([v],M)~:~Mv=0\}$. Indeed $R(X)=\Delta$ and $X$ is smooth, for the differential of the map $$ \begin{eqnarray} F~:~\mathbb R^n\times M_n(\mathbb R) \longrightarrow & \mathbb R^n \\ (v,M)\longmapsto & Mv \end{eqnarray} $$ is $(h,A)\mapsto Mh+Av$, which has maximal rank $n$ if $v\neq0$ (take $h:=0$).

Answer 2

In the paper

• Edward Bierstone: Differentiable functions, BOLL. SOC. BRAS. MAT.,VOL 11 N 2 (1980), 139-190

one finds the following theorem, which states that for subanalytic subsets (those given by equations and $\le$-inequalities involving real analytic functions) with dense interior the interior distance is locally Hoelder continuous with respect to Euclidean distance.

Theorem 6.17. Let $V$ be an open subset of $\mathbb R^n$, and $A$ a closed subanalytic subset of $V$ such that $Int(A)$ is dense in $A$. For every compact subset $L$ of $A$, there exists $c > 0$ and an integer $\alpha \ge 1$ such that any two points $b, y \in L$ can be joined by a semianalytic arc $\sigma$ in $A$ such that:

• (1) $|\sigma| < c|b - y|^{1/\alpha}$

• (2) $\sigma$ intersects $\partial A$ in at most finitely many points.

Answer 3

This applies to the earlier version of the question:

It is an open subset since the determinant function is continuous, and open subsets are locally path-connected.

Answer 4

One way is to reduce first to $x,y$ with $n$ distinct complex eigenvalues. Then the $n$ eigenvalues and eigenspaces of x are close to those of y. Now connect all those with paths (just make sure not to run through eigenvalue zero, and to keep invariance under complex conjugation).