A complete locally CAT(0) circle whose fundamental group prevents global CAT(0)

(i) is the first part of [F1]: $S_{\mathrm{\ell }}^1$ is compact, complete and geodesic, hence a length space, and it is locally isometric to $\mathbb{R}$.

Small balls are intervals. Let $x\in S_{\mathrm{\ell }}^1$, $0<r<\mathrm{\ell }\mathrm{/}4$ and $t$ any lift of $x$. For $s,s^{\mathrm{\prime }}\in [t-r,t+r]$ we have $\mathrm{\mid }s-s^{\mathrm{\prime }}\mathrm{\mid }\le 2r<\mathrm{\ell }\mathrm{/}2$, so $d_{\mathrm{\ell }}(p(s),p(s^{\mathrm{\prime }}))=\min \{\mathrm{\mid }s-s^{\mathrm{\prime }}\mathrm{\mid },\mathrm{\ell }-\mathrm{\mid }s-s^{\mathrm{\prime }}\mathrm{\mid }\}=\mathrm{\mid }s-s^{\mathrm{\prime }}\mathrm{\mid }$; hence $p$ restricts to a distance-preserving bijection of the interval $[t-r,t+r]$ onto the closed ball $\bar{B}(x,r)$.

An interval is CAT(0). Let $V=[a,b]\subset \mathbb{R}$ with the induced metric; it is geodesic, and a geodesic triangle with vertices $\alpha \le \beta \le \gamma$ in $V$ has its three sides contained in $[\alpha ,\gamma ]$ and side lengths $\beta -\alpha ,\gamma -\beta ,\gamma -\alpha$, so its Euclidean comparison triangle is the degenerate segment $[\bar{\alpha },\bar{\gamma }]$ of length $\gamma -\alpha$ with the three comparison vertices at positions $\alpha ,\beta ,\gamma$; a point of the triangle lies on some side and its comparison point has the same position in $[\bar{\alpha },\bar{\gamma }]$, so all distances between points of the triangle equal their comparison distances and the CAT(0) inequality holds with equality. Hence $V$ is CAT(0).

Conclusion of (i). By steps 1.2 and 1.3 each point of $S_{\mathrm{\ell }}^1$ has a closed ball of radius $r<\mathrm{\ell }\mathrm{/}4$ that is isometric, for the induced metric, to an interval, and intervals are CAT(0); so $S_{\mathrm{\ell }}^1$ is locally CAT(0).

(ii) $p$ is a covering map. The quotient map $p$ is continuous and surjective, and for every $x$ and $0<r<\mathrm{\ell }\mathrm{/}4$ the preimage $p^{-1}(B(x,r))$ is the disjoint union of the open intervals $(t_k-r,t_k+r)$ about the lifts $t_k=t+k\mathrm{\ell }$ of $x$, since two such intervals meet only if their centres differ by less than $2r<\mathrm{\ell }$; by step 1.2 each of them is mapped isometrically onto $B(x,r)$. Hence every ball of radius $<\mathrm{\ell }\mathrm{/}4$ is evenly covered and $p$ is a covering map.

(iii) the witness triple. Let $a:=0$, $b:=\mathrm{\ell }\mathrm{/}3$, $c:=2\mathrm{\ell }\mathrm{/}3$ and $m:=\mathrm{\ell }\mathrm{/}6$ in $S_{\mathrm{\ell }}^1$. The three pairwise distances are $d_{\mathrm{\ell }}(a,b)=d_{\mathrm{\ell }}(b,c)=d_{\mathrm{\ell }}(a,c)=\mathrm{\ell }\mathrm{/}3$, since $\mathrm{\mid }0-\mathrm{\ell }\mathrm{/}3\mathrm{\mid }=\mathrm{\mid }\mathrm{\ell }\mathrm{/}3-2\mathrm{\ell }\mathrm{/}3\mathrm{\mid }=\mathrm{\ell }\mathrm{/}3$ and $d_{\mathrm{\ell }}(a,c)=\min \{2\mathrm{\ell }\mathrm{/}3,\mathrm{\ell }-2\mathrm{\ell }\mathrm{/}3\}=\mathrm{\ell }\mathrm{/}3$; moreover $m$ lies on the geodesic $[a,b]$ given by the arc from $0$ to $\mathrm{\ell }\mathrm{/}3$, and $d_{\mathrm{\ell }}(m,c)=\min \{\mathrm{\ell }\mathrm{/}2,\mathrm{\ell }-\mathrm{\ell }\mathrm{/}2\}=\mathrm{\ell }\mathrm{/}2$, so there is a geodesic triangle of $S_{\mathrm{\ell }}^1$ whose comparison is tested.

(ii) The generator is not nullhomotopic. The loop $\alpha (t)=p(t)$, $t\in [0,\mathrm{\ell }]$, lifts from $0$ to $\tilde{\alpha }(t)=t$ and ends at $\mathrm{\ell }$, whereas the based constant loop lifts to a path ending at $0$. Thus [F4] excludes a based nullhomotopy, giving a nontrivial class in ${\pi }_1(S_{\mathrm{\ell }}^1,0)$. To exclude a free nullhomotopy as well, suppose $K:[0,\mathrm{\ell }]\times [0,1]\to S_{\mathrm{\ell }}^1$ deforms $\alpha$ through loops to a constant loop; then $K(0,s)=K(\mathrm{\ell },s)$. Lift $K$ with initial lift $t\mapsto t$. The two paths $s\mapsto \tilde{K}(\mathrm{\ell },s)$ and $s\mapsto \tilde{K}(0,s)+\mathrm{\ell }$ lift the same path and both start at $\mathrm{\ell }$, hence coincide by [F4]. At $s=1$ the lifted constant loop is constant by path-lifting uniqueness in [F4]: the constant path at its initial lift is another lift of the same constant loop. Their difference is then both $\mathrm{\ell }$ and $0$, impossible. The circle is path-connected by its arcs and is not simply connected.

(iii) the comparison fails. The Euclidean comparison triangle of $(a,b,c)$ is equilateral of side $\mathrm{\ell }\mathrm{/}3$, and the comparison point $\bar{m}$ of $m$ is the midpoint of the side $[\bar{a},\bar{b}]$; its distance to the opposite vertex $\bar{c}$ is the altitude $\mathrm{\ell }\sqrt{3}\mathrm{/}6$, because $(\mathrm{\ell }\sqrt{3}\mathrm{/}6{)}^2+(\mathrm{\ell }\mathrm{/}6{)}^2={\mathrm{\ell }}^2\mathrm{/}9=(\mathrm{\ell }\mathrm{/}3{)}^2$ by Pythagoras in ${\mathbb{E}}^2$. Since $\mathrm{\ell }\sqrt{3}\mathrm{/}6<\mathrm{\ell }\mathrm{/}2$ we have $d_{\mathrm{\ell }}(m,c)=\mathrm{\ell }\mathrm{/}2>d_2(\bar{m},\bar{c})$, so the CAT(0) inequality fails and $S_{\mathrm{\ell }}^1$ is not CAT(0).

The fundamental group is infinite cyclic. Parametrize based loops on $[0,1]$. Every loop $\beta$ lifts uniquely from $0$ to a path $\tilde{\beta }$ in $\mathbb{R}$, with endpoint $n\mathrm{\ell }$ for a unique integer $n$; [F4] makes $n$ invariant under based homotopy. Conversely $p((1-s)\tilde{\beta }(t)+sn\mathrm{\ell }t)$ is a based homotopy to the loop $t\mapsto p(n\mathrm{\ell }t)$, since the two endpoints of the interpolated lift stay $0,n\mathrm{\ell }$. Every integer is realized by this explicit loop. When loops of winding $n,m$ are concatenated, the lift of the second starts at $n\mathrm{\ell }$ and is its lift from $0$ translated by $n\mathrm{\ell }$, so the endpoint is $(n+m)\mathrm{\ell }$. Winding thus gives an isomorphism ${\pi }_1(S_{\mathrm{\ell }}^1,0)\cong \mathbb{Z}$, sending $\alpha$ to $1$.

(iv). Steps 1.1–1.3 and 2.1 show that $S_{\mathrm{\ell }}^1$ is a complete locally CAT(0) length space, path-connected because it is geodesic and hence connected, while step 3.1 shows that it is not simply connected and step 3.2 that it is not CAT(0); hence the simple-connectivity hypothesis of the globalization theorem [F6] cannot be dropped.

The circle is not contractible, and the theorem's clauses fail explicitly. A contraction of the circle, composed with $\alpha$, would be a free nullhomotopy of $\alpha$, excluded by step 3.1. Thus the conclusion of clause (iii) of [F6] fails, and its clause (i) fails as well: the points $0$ and $\mathrm{\ell }\mathrm{/}2$ are joined by exactly two minimizing geodesics, the two semicircular arcs of length $\mathrm{\ell }\mathrm{/}2$. Indeed lift any minimizing segment on $[0,1]$ from $0$ through $p$: on each interval chart its lift is affine with slope either $\mathrm{\ell }\mathrm{/}2$ or $-\mathrm{\ell }\mathrm{/}2$, and the slope cannot change on overlapping intervals, so the lift is precisely $t\mapsto \pm t\mathrm{\ell }\mathrm{/}2$. Thus the unique geodesic from $x_0$ to $x$ that builds the geodesic contraction is not available at $x=\mathrm{\ell }\mathrm{/}2$ and $0<t<1$.