Homeomorphism

A one-to-one correspondence between two topological spaces such that the two mutually-inverse mappings defined by this correspondence are continuous. These mappings are said to be homeomorphic, or topological, mappings, and also homeomorphisms, while the spaces are said to belong to the same topological type or are said to be homeomorphic or topologically equivalent. They are isomorphic objects in the category of topological spaces and continuous mappings. A homeomorphism must not be confused with a condensation (a bijective continuous mapping); however, a condensation of a compactum onto a Hausdorff space is a homeomorphism.

Examples. 1) The function $1/(e^X+1)$ establishes a homeomorphism between the real line $\mathbb{R}$ and the interval $(0,1)$; 2) a closed circle is homeomorphic to any closed convex polygon; 3) three-dimensional projective space is homeomorphic to the group of rotations of the space $\mathbb{R}^3$ around the origin and also to the space of unit tangent vectors to the sphere $S^2$; 4) all compact zero-dimensional groups with a countable base are homeomorphic to the Cantor set; 5) all infinite-dimensional separable Banach spaces, and even all Fréchet spaces, are homeomorphic; 6) a sphere and a torus are not homeomorphic.

The term "homeomorphism" was introduced in 1895 by H. Poincaré [3], who applied it to (piecewise-) differentiable mappings of domains and submanifolds in $\mathbb{R}^n$; however, the concept was known earlier, e.g. to F. Klein (1872) and, in a rudimentary form, to A. Möbius (as an elementary likeness, 1863). At the beginning of the 20th century homeomorphisms began to be studied without assuming differentiability, as a result of the development of set theory and the axiomatic method. This problem, which was explicitly stated for the first time by D. Hilbert [7], forms the content of Hilbert's fifth problem. Of special importance was the discovery by L.E.J. Brouwer that $\mathbb{R}^n$ and $\mathbb{R}^m$ are not homeomorphic if $n \neq m$. This discovery restored the faith put by mathematicians in geometric intuition. This faith had been shaken by G. Cantor's result stating that $\mathbb{R}^n$ and $\mathbb{R}^m$ have the same cardinality and by the result obtained by G. Peano on the possibility of a continuous mapping from $\mathbb{R}^n$ onto $\mathbb{R}^m$, $n < m$. The concepts of a metric (or, respectively, a topological) space, introduced by M. Fréchet and F. Hausdorff, laid a firm foundation for the concept of a homeomorphism and made it possible to formulate the concepts of a topological property (a property which remains unchanged under a homeomorphism), of topological invariance, etc., and to formulate the problem of classifying topological spaces of various types up to a homeomorphism. However, when presented in this manner, the problem becomes exceedingly complicated even for very narrow classes of spaces. In addition to the classical case of two-dimensional manifolds, such a classification was given only for certain types of graphs, for two-dimensional polyhedra and for certain classes of manifolds. The general problem of classification cannot be algorithmically solved at all, since it is impossible to obtain an algorithm for distinguishing, say, manifolds of dimension larger than three. Accordingly, the classification problem is usually posed in the framework of a weaker equivalence relation, e.g. in algebraic topology using homotopy type or, alternatively, to classify spaces having a certain specified structure. Even so, the homeomorphism problem remains highly important. In the topology of manifolds it was only in the late 1960s that methods for studying manifolds up to a homeomorphism were developed. These studies are carried out in close connection with homotopic, topological, piecewise-linear, and smooth structures.

A second problem is the topological characterization of individual spaces and classes of spaces (i.e. a specification of their characteristic topological properties, formulated in the language of general topology, cf. Topology, general and Topological invariant). This has been solved, for example, for one-dimensional manifolds, two-dimensional manifolds, Cantor sets, the Sierpiński curve, the Menger curve, pseudo-arcs, Baire spaces, etc. Spectra furnish a universal tool for the topological characterization of spaces; Aleksandrov's homeomorphism theorem was obtained using spectra [4]. The sphere and, in general, the class of locally Euclidean spaces, has been characterized by a sequence of subdivisions gradually diminishing in size [5]. A description of locally compact Hausdorff groups by means of spectra has been given [6]. Another method is to consider various algebraic structures connected with the mappings. Thus, a compact Hausdorff space is homeomorphic to the space of maximal ideals of the algebra of real functions defined on it. Many spaces are characterized by the semi-group of continuous mappings into themselves (cf. Homeomorphism group). In general topology a topological description is given of numerous classes of topological spaces. The characterization of spaces inside a given class is also of interest. Thus, it is very useful to describe a sphere as a compact manifold covered by two open cells. The problem of algorithmic identification of spaces has not been studied much. At the time of writing (1977) it has not been solved for the sphere $S^n$ where $n \ge 3$.

In general, the non-homeomorphism of two topological spaces is proved by specifying a topological property displayed by only one of them (compactness, connectedness, etc.; e.g., a segment differs from a circle in that it can be divided into two by one point); the method of invariants is especially significant in this connection. Invariants are either defined in an axiomatic manner for a whole class of spaces at the same time, or else algorithmically, according to a specific representation of the space, e.g. by triangulation, by the Heegaard diagram, by decomposition into handles (cf. Handle theory), etc. The problem in the former case is to compute the invariant, while in the latter it is to prove topological invariance. An intermediate case is also possible — e.g. characteristic classes (cf. Characteristic class) of smooth manifolds were at first defined as obstructions to the construction of vector and frame fields, and later as the image of the tangent bundle under mappings of the $KO$-functor into a cohomology functor, but in neither case can the respective problems be solved by definition. Historically the first example of proving topological invariance (of the linear dimension of $\mathbb{R}^n$) was given by Brouwer in 1912. The classical method, due to Poincaré, is to begin by giving both definitions — the "computable" and the "invariant" — and then to prove that they are identical. This method proved especially useful in the theory of homology of a polyhedron. Another method is to prove that an invariant remains unchanged under elementary transformations of a representation of the space (e.g. subdivision by triangulations). It is completed if it is known that it is possible to obtain all the representations of a given type in this manner. Thus, the so-called "Hauptvermutung" of combinatorial topology arose in the topology of polyhedra in this connection. This method (which was also proposed by Poincaré) proved highly useful in the topology of two and three dimensions, in particular in knot theory, but it is out of use now (except for the constructive direction) not so much because the "Hauptvermutung" proved to be untrue, as because the development of category theory made it possible to give more realistic definitions, more in accordance with the subject matter, with a more accurate presentation of the problem of computation and topological invariance. Thus, the invariance of homology groups, which are defined functorially for spaces but are defined in a computable manner for complexes, follows from the comparison of the category of complexes and homotopy classes of simplicial mappings with the category of homotopy classes of continuous mappings. In this way one does not have to give a separate definition for a large category and one can extend it to a smaller category as well. (The sources of this idea are found in Brouwer's theory of degree.) The superiority of the new method was seen to be particularly evident in connection with the second definition of characteristic classes, given above, as transformations of functors. Thus, the problem of topological invariance naturally turned out to be a part of the question of the relation between the $K$-functor and its topological generalization.

If two spaces are homeomorphic, then the method of spectra (and of diminishing subdivisions) is the only one of general value for the establishment of homeomorphism. On the other hand, if a classification has already been constructed the problem is solved by comparison of invariants. In practice the establishment of homeomorphism often proves to be a very difficult geometrical problem, which must be solved by employing special tools. Thus, homeomorphism of Euclidean spaces and some of their quotient spaces is established using a pseudo-isotopy.

References

[1]	D. Hilbert, S.E. Cohn-Vossen, "Anschauliche Geometrie" , Springer (1932)
[2]	V.G. Boltyanskii, V.A. Efremovich, "Outline of new ideas in topology" Mat. Prosveshchenie , 2 (1957) pp. 3–34 (In Russian)
[3]	H. Poincaré, "Oeuvres" , 2 , Gauthier-Villars (1952)
[4]	P.S. Aleksandrov, "Topological duality theorems. Part 2. Non-closed sets" Trudy Mat. Inst. Steklov. , 54 (1959) (In Russian)
[5]	O.G. Harrold, "A characterization of locally Euclidean spaces" Trans. Amer. Math. Soc. , 118 (1965) pp. 1–16
[6]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[7]	"Hilbert problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 437–479 (Translated from German)