In this talk, I would like to focus on wall singularity of metric spaces with an upper curvature bound. Lytchak and I have studied basic geometric structure of GCBA spaces. A GCBA space means a locally compact, separable, locally geodesically complete metric space with an upper curvature bound. I will report on a wall singularity theorem and a regularity theorem of codimension two for GCBA spaces.

Let $T_k^0$ be the discrete metric space consisting of $k$ points with pairwise distance $\pi$, and $T_k^1$ the Euclidean cone over $T_k^0$. The $\ell_2$-product metric space $\mathbb{R} \times T_k^1$ has wall singularity along a line, provided $k \ge 3$.

Let $X$ be a GCBA space. We say that a point $x \in X$ is an $n$-wall point in $X$ if the tangent space $T_xX$ at $x$ in $X$ isometrically splits as $\mathbb{R}^{n-1} \times T_k^1$ for some $k \ge 3$. We denote by $W_n(X)$ the set of all $n$-wall points in $X$, and call it the $n$-wall singular set in $X$.

As a wall singularity theorem, we conclude that for every $n$-wall point $x \in W_n(X)$ in $X$, and for every open neighborhood $U$ of $x$, we can find a point $x_0 \in U$ arbitrarily close to $x$, and an open neighborhood $U_0$ of $x_0$ contained in $U$, such that $U_0$ is homeomorphic to $\mathbb{R}^{n-1} \times T_{k_0}^1$ for some $k_0 \ge 3$; moreover, $\mathcal{H}^{n-1}(S(U_0))$ is positive and finite, where $\mathcal{H}^{n-1}$ is the $(n-1)$-dimensional Hausdorff measure, and $S(U_0)$ is the set of all non-manifold points in $U_0$.

As a regularity theorem of codimension two, we establish that for every purely $n$-dimensional open subset $U$ of $X$ the following are equivalent: (1) the $n$-wall singular set $W_n(U)$ in $U$ is empty; (2) there exists no open subset $U_0$ contained in $U$ such that $U_0$ is homeomorphic to $\mathbb{R}^{n-1} \times T_{k_0}^1$ for some $k_0 \ge 3$; (3) $\dim S(U) \le n-2$, where $\dim$ is the topological dimension; (4) $\dim_{\mathrm{H}} S(U) \le n-2$, where $\dim_{\mathrm{H}}$ is the Hausdorff dimension.