In
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...
(a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
), the Whitehead theorem states that if a
continuous mapping
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
''f'' between
CW complex
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
es ''X'' and ''Y'' induces
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
s on all
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s, then ''f'' is a
homotopy equivalence
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
. This result was proved by
J. H. C. Whitehead
John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as "Henry", was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princet ...
in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping.
Statement
In more detail, let ''X'' and ''Y'' be
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s. Given a continuous mapping
:
and a point ''x'' in ''X'', consider for any ''n'' ≥ 0 the induced
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
:
where π
''n''(''X'',''x'') denotes the ''n''-th homotopy group of ''X'' with base point ''x''. (For ''n'' = 0, π
0(''X'') just means the set of
path component
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties th ...
s of ''X''.) A map ''f'' is a weak homotopy equivalence if the function
:
is
bijective
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
, and the homomorphisms
are bijective for all ''x'' in ''X'' and all ''n'' ≥ 1. (For ''X'' and ''Y''
path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
, the first condition is automatic, and it suffices to state the second condition for a single point ''x'' in ''X''.) The Whitehead theorem states that a weak homotopy equivalence from one CW complex to another is a homotopy equivalence. (That is, the map ''f'': ''X'' → ''Y'' has a homotopy inverse ''g'': ''Y'' → ''X'', which is not at all clear from the assumptions.) This implies the same conclusion for spaces ''X'' and ''Y'' that are homotopy equivalent to CW complexes.
Combining this with the
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results ...
yields a useful corollary: a continuous map
between
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
CW complexes that induces an isomorphism on all integral
homology groups is a homotopy equivalence.
Spaces with isomorphic homotopy groups may not be homotopy equivalent
A word of caution: it is not enough to assume π
''n''(''X'') is isomorphic to π
''n''(''Y'') for each ''n'' in order to conclude that ''X'' and ''Y'' are homotopy equivalent. One really needs a map ''f'' : ''X'' → ''Y'' inducing an isomorphism on homotopy groups. For instance, take ''X''=
''S''2 ×
RP3 and ''Y''= RP
2 × ''S''
3. Then ''X'' and ''Y'' have the same
fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
, namely the
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
Z/2, and the same universal cover, namely ''S''
2 × ''S''
3; thus, they have isomorphic homotopy groups. On the other hand their homology groups are different (as can be seen from the
Künneth formula Künneth is a surname. Notable people with the surname include:
* Hermann Künneth (1892–1975), German mathematician
* Walter Künneth (1901–1997), German Protestant theologian
{{DEFAULTSORT:Kunneth
German-language surnames ...
); thus, ''X'' and ''Y'' are not homotopy equivalent.
The Whitehead theorem does not hold for general topological spaces or even for all subspaces of R
n. For example, the
Warsaw circle
Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory ...
, a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
subset of the plane, has all homotopy groups zero, but the map from the Warsaw circle to a single point is not a homotopy equivalence. The study of possible generalizations of Whitehead's theorem to more general spaces is part of the subject of
shape theory.
Generalization to model categories
In any
model category
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
, a weak equivalence between cofibrant-fibrant objects is a homotopy equivalence.
References
* J. H. C. Whitehead, ''Combinatorial homotopy. I.'', Bull. Amer. Math. Soc., 55 (1949), 213–245
* J. H. C. Whitehead, ''Combinatorial homotopy. II.'', Bull. Amer. Math. Soc., 55 (1949), 453–496
* A. Hatcher
''Algebraic topology'' Cambridge University Press, Cambridge, 2002. xii+544 pp. and {{isbn, 0-521-79540-0 (see Theorem 4.5)
Theorems in homotopy theory