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.

Asymptotic Ouroboros

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<u. 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 hormos — topological semigroup

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