Alternating Cone Spaces

The Altenating Cone Space is a space that illustrates strange phenomenon in homotopy theory. In particular, the alternating cone over the circle is a simply connected, cell-like space where non-intuitive higher homotopy/homology pops up. The construction involves attaching cones of shrinking diameter to a square in alternating direction so that they limit on an edge of the square. It was introduced by Eda, Karmiov, and Repovs in [3].

The Alternating Cone Space (over the circle)

What is most striking about the space AC(S^1) is that it is cell-like (anything thickening of it is contractible) and simply connected and yet \pi_2(AC(S^1)\cong H_2(AC(S^1)) is uncountable even though there does not appear to be an “enclosed space” or “twists” to create non-trivial second homology.

There is also a more genreal construction studied in [3]. Note that the cone CX=X\times [0,1]/X\times\{1\} over any space X can be used in place of the cone over S^1. The gluing is done along the spine \{x\}\times [0,1] of each cone for some given baspepoint x\in X. This results in a functor (X,x)\mapsto AC(X,x) on based spaces. When the basepoint is clear from context, it makes sense to write AC(X) (in fact the homotopy type of AC(X,x) is independent of the choice of basepoint of X).

Constructions similar with alternating cones, e.g. “snake cone” SC(S^1) have been studied in [1,2,4]. It is worth noting that it is shown in Theorem 1.2 of [4] that AC(S^1) and SC(S^1) are homotopy equivalent. While snake cones were constructed first, alternating cones are significantly easier to work with.

Geometric Construction: Start with the unit square S=[0,1]^2\times \{0\} in 3-space. For odd n\geq 1, let C_n be the cylindrical cone

\{\left(\frac{s \cos(2 \pi t)}{10n}+\frac{1}{m},s, \frac{s(\sin(2\pi t)+1)}{10n}\right)\in\mathbb{R}^3\mid 0\leq s,t\leq 1\}.

For even n\geq 2, let C_n be the cylindrical cone

\{\left(\frac{(1-s) \cos(2 \pi t)}{10n}+\frac{1}{m},s, \frac{(1-s)(\sin(2\pi t)+1)}{10n}\right)\in\mathbb{R}^3\mid 0\leq s,t\leq 1\}.

The Alternating Cone over the circle is AC(S^1)=S\cup \bigcup_{n\geq 1}C_n.

Topological Properties: 2-dimensional Peano continuum (path-connected, locally path-connected, compact metric space). AC(S^1) “cell-like” since any \epsilon-thickening is contractible. However, the space itself is not contractible.

Fundamental Group: AC(X) is simply connected for any based Peano continuum X (see Theorem 3.2 of [3]).

Homology groups: While it is known that H_1(AC(S^1))=0 and H_2(AC(S^1)) is non-trivial (see Theorem 3.5 of [3]), not much else is known about these groups.

Higher homotopy groups: \pi_2(AC(S^1)) is known to be non-trivial (see Theorem 3.5 of [3]) but nothing seems to be known about the groups \pi_n(AC(S^1)), n\geq 3.

Cech Homotopy groups: \check{\pi}_n(AC(S^1))=0 for all n.

Cech (co)homology groups: \check{H}_n(AC(S^1))=\check{H}^n(AC(S^1))=0 for all n.

Other Properties:

  • Semi-locally simply connected: Yes
  • Locally simply connected: No
  • Traditional Universal Covering Space: Yes, itself
  • Generalized Universal Covering Space: Yes, itself
  • Homotopically Hausdorff: Yes, trivially
  • Strongly (freely) Homotopically Hausdorff: Yes, trivially
  • Homotopically Path-Hausdorff: Yes, trivially
  • 1–UV_0: No, small inessential loops may require homotopies with large diameters to contract.
  • \pi_1-shape injective: Yes, trivially

References

[1] K. Eda, U. H. Karimov, D. Repovs, A construction of simply connected noncontractible cell-like two-dimensional continua, Fundamenta Math. 195 (2007), 193-203.

[2] K. Eda, U. Karimov, D. Repovˇs, On the singular homology of one class of simply-connected cell-like spaces, Mediterranean J. Math. 8:2 (2011), 153–160.

[3] K. Eda, U. H. Karimov, D. Repovs, On 2-dimensional nonaspherical cell-like Peano continua. A simple approach, Mediterranean J. Math. 10 (2013), 519–528.

[4] K. Eda, U. H. Karimov, D. Repovs, A. Zastrow, On Snake Cones, Alternating Cones, and Related Constructions, Glasnik Mat. 48 (2013), no. 1, 115-135.