# Blue-Sky Catastrophe

The blue-sky catastrophe is a space, which arises from the bifurcation of a certain dynamical system. This space may be described as the closure of a certain embedding of a cylinder $S^1\times (0,1)$ in $\mathbb{R}^3$. This space can be described by starting with $S^1\times [0,\infty)$ and allowing the clinder to circle back on $S^1\times\{0\}$, winding around towards it infinitely many times.

This space is topologically interesting because of the combination of which properties it does and does not have.

The Blue-Sky Catastrophe space $B$ can be formally described as follows: Let $\epsilon:[0,\infty)\to S^1$, $\epsilon(t)=(\cos(2\pi t),\sin(2\pi t))$ be the exponential map. Let $(a,b)$ be an open arc in $S^1$ of diameter less than $1/2$ (the choice of $1/2$ is arbitrary, we just want to consider small intervals). For a natural number $N$, let $U(a,b,N)\subseteq [0,\infty)$ be the union of the connected components of $\epsilon^{-1}((a,b))$, which are contained in $[N,\infty)$. We define the blue-sky catastrophe space $B$ to have underlying set $S^1\times [0,\infty)$ but with a topology coarser than the usual product topology. A basic neighborhood at a point $(t,u)$ for $u>0$ is a “standard one” of the form $(a,b)\times (u-\epsilon,u+\epsilon)$, $0<\epsilon. More interestingly, a basic neighborhood of a point $(t,0)$ is a set of the form

$((a,b)\times [0,\epsilon))\cup (S^1\times U(N,a,b))$

for an open arc $(a,b)$ of $t$ of diameter $<1/2$, $\epsilon>0$, and natural number $N\geq 1$. Such a set looks like a square with only one boundary edge and a sequence of open cylinders of shrinking diameter approaching that boundary edge. The use of the exponential map is ensures the basic neighborhoods actually describe a coherent “winding” toward the limiting circle $S^1\times\{0\}$.

Alternative names: Hyperbolic smile

Topological Properties:  2-dimensional compact metric space. Embeds in $\mathbb{R}^3$. If you remove the limiting circle, the result $B\backslash S^1\times \{0\}$ is homeomorphic to an open annulus. This space is not locally path-connected at any point in the “limiting circle” $S^1\times\{0\}$.

Homotopy and homology groups: If $S^1\times [0,\infty)$ has the usual product topology, then the identity function $S^1\times [0,\infty)\to B$ is continuous and, moreover, is a weak homotopy equivalence. In particular, $\pi_1(B)\cong H_1(B)\cong \mathbb{Z}$ is generated by a loop going around the limiting circle. Any compact subset of $B$ must be contained in $S^1\times [0,r)$ for some $r$ so no loop can go along the entire length of the winding surface. All higher homotopy and homology groups are trivial.

Homotopy Type and Shape Type: Blue sky catastrophe is shape equivalent to the unit circle. When one “thickens” $B$ (embedding as above in $\mathbb{R}^3$) in the construction above by any $\epsilon >0$, the resulting space is homotopy equivalent to a circle. It follows that $W$ is not contractible since it is not shape equivalent to a point.

Cech Homotopy groups: $\check{\pi}_n(B)\cong \begin{cases} \mathbb{Z}, & n=1 \\ 0, & n \neq 1 \end{cases}$.

Interetsingly, the groups $\pi_1(B)$ and $\check{\pi}_1(B)$ describe completely different things even though they are both isomorphic to $\mathbb{Z}$. This coincidence is purely a result of using $S^1$ instead of some other manifold. Indeed, $\pi_1(B)$ is generated by an actual loop going around the limiting circle while $\check{\pi}_1(B)$ is generated not by an actual loop but by a loop going around the entire structure once $B$ is thickened by some small $\epsilon$.

Cech Homology groups: $\check{H}_n(W)\cong \begin{cases} \mathbb{Z}, & n=1,0 \\ 0, & n >1 \end{cases}$

Wild Set: The space $B$ is semilocally simply connected in the based sense (for every $x\in B$, there exists a neighborhood $U$ of $x$ such that the homomorphism $\pi_1(U,x)\to \pi_1(B,x)$ induced by inclusion is trivial). However, at the points of the limiting circle, $B$ is not semilocally simply connected in the unbased sense (for every $x\in B$, there exists a neighborhood $U$ of $x$ such that, for every $y\in U$, the homomorphism $\pi_1(U,y)\to \pi_1(B,y)$ induced by inclusion is trivial). So it depends what kind of wild set you’re defining. With the usual definition, $B$ has no wild set. But if you define $\mathbf{w}_{ub}(X)$ to be the subspace of points at which $X$ is not semilocally simply connected in the unbased sense, then $\mathbf{w}_{ub}(B)=S^1\times\{0\}$.

Covering spaces: The only covering map $p:E\to B$ where $E$ is path-connected is the identity map. Hence, $B$ has no non-trivial path-connected covering spaces. However, $B$ does have non-trivial connected covering spaces. In particular, there is a unique covering space corresponding to each subgroup of $\check{\pi}_1(B)\cong\mathbb{Z}$ in a way that is reminscent of the covering spaces of $S^1$. In particular the trivial subgroup $1\leq \mathbb{Z}$ corresponds to the universal covering over $B$ that looks like this:

The universal connected cover of $B$ (the two limiting points in the picture are not included). Image from: “The structure of topological semigroups — revisited”, Paul S. Mostert. Bull. AMS 1966.

Summary and Other Properties:

• Semi-locally simply connected (in the based sense): Yes
• Semi-locally simply connected (in the based sense): Yes
• Traditional Universal Covering Space: No, there is a universal connected covering space but not a simply connected path-connected covering space
• Generalized Universal Covering Space: Yes
• Homotopically Hausdorff: Yes
• Strongly (freely) Homotopically Hausdorff: No
• Homotopically Path-Hausdorff: Yes
• $1-UV_0$Yes
• $\pi_1$-shape injective: No