Replies

Alexander's Horned Sphere

The above topological structure, composed of a countable union of compact sets, is called Alexander's horned sphere. It is homeomorphic with the ball , and its boundary is therefore a sphere. It is therefore an example of a wild embedding in . The outer complement of the solid is not simply connected, and its fundamental group is not finitely generated. Furthermore, the set of nonlocally flat ("bad") points of Alexander's horned sphere is a Cantor set.

The horned sphere as originally drawn by Alexander (1924) is illustrated above.

The complement in of the bad points for Alexander's horned sphere is simply connected, making it inequivalent to Antoine's horned sphere. Alexander's horned sphere has an uncountable infinity of wild points, which are the limits of the sequences of the horned sphere's branch points (roughly, the "ends" of the horns), since any neighborhood of a limit contains a horned complex.

A humorous drawing by Simon Frazer (Guy 1983, Schroeder 1991, Albers 1994) depicts mathematician John H. Conway with Alexander's horned sphere growing from his head.

Antoine's Horned Sphere

Albers, D. J. Illustration accompanying "The Game of 'Life."' Math Horizons, p. 9, Spring 1994.

Alexander, J. W. "An Example of a Simply Connected Surface Bounding a Region Which Is Not Simply Connected." Proc. N. A. S. 10, 8-10, 1924.

Guy, R. "Conway's Prime Producing Machine." Math. Mag. 56, 26-33, 1983.

Hocking, J. G. and Young, G. S. Topology. New York: Dover, 1988.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 80-81, 1976.

Schroeder, M. Fractals, Chaos, Power Law: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 58, 1991.

Throwing in the kitchen ~~sink~~ horned sphere, eh?

Thank you for the links and information on Alexander's Horned Sphere! As I understand it, the manifolds in ordinary string theory are viewed as smooth. However, wrt topology and string theory - you might find this article interesting (emphasis mine):

CERN Courier - Space goes quantum at Stony - IOP Publishing - article

... The new picture that emerges from this duality is that of a "quantum" CY geometry. To understand what this means it is worth recalling that in a classical space of any kind each point is specified by a set of numbers, or co-ordinates. Examples of co-ordinates are the longitude and latitude of the Earth's surface. In the quantum CY space the co-ordinates are no longer simple numbers to be specified at will. Rather they obey the Heisenberg uncertainty principle, which relates the position and momentum of a quantum particle. For the quantum CY spaces of Okounkov, Reshetikhin and Vafa's dual description of topological string theory, the long-standing dream of replacing a smooth classical space with a discrete quantum substructure is thus realized. In this system the emergence of a classical geometry out of a quantum system can be clearly controlled and understood. As is shown in further work by Vafa et al., this gives an explicit and controllable picture of the Wheeler-Hawking notion of topological fluctuations - or "foam" - in space-time (Iqbal et al. 2003). The fluctuations of topology and geometry actually become the deep origin of strings. They extend rather than reduce the predictive power of the quantum theory of gravity.

Of course many challenges remain before a full theory of this kind can be realized. Chief among these is the extension of the picture from topological strings to full string theory. A possible path has been identified, however, suggesting that in string theory, as in Einstein's gravity, the distinction between forces and the space in which they act melts away.

I strongly suspect the last statement is the key and will prevail upon further scrutiny, i.e. Einstein's dream of transmuting the base wood of matter into the pure marble of geometry.