Homotopic

Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. For example, the real line is homotopic to a single point, as is any tree. However, the circle is not contractible, but is homotopic to a solid torus. The basic version of homotopy is between maps. Two maps  and  are homotopic if there is a continuous map

such that  and .

Whether or not two subsets are homotopic depends on the ambient space. For example, in the plane, the unit circle is homotopic to a point, but not in the punctured plane . The puncture can be thought of as an obstacle.

However, there is a way to compare two spaces via homotopy without ambient spaces. Two spaces  and  are homotopy equivalent if there are maps  and  such that the composition  is homotopic to the identity map of  and  is homotopic to the identity map of . For example, the circle is not homotopic to a point, for then the constant map would be homotopic to the identity map of a circle, which is impossible because they have different Brouwer degrees.

REFERENCES:

Aubry, M. Homotopy Theory and Models. Boston, MA: Birkhäuser, 1995.

Collins, G. P. "The Shapes of Space." Sci. Amer. 291, 94-103, July 2004.

Krantz, S. G. "The Concept of Homotopy" §10.3.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 132-133, 1999.