In
homotopy theory (a branch of
mathematics), the Whitehead theorem states that if a
continuous mapping
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
''f'' between
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
es ''X'' and ''Y'' induces
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
s on all
homotopy groups, then ''f'' is a
homotopy equivalence
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
. This result was proved by
J. H. C. Whitehead 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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, 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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s. Given a continuous mapping
:
and a point ''x'' in ''X'', consider for any ''n'' ≥ 1 the induced
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
:
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 ...
s of ''X''.) A map ''f'' is a weak homotopy equivalence if the function
:
is
bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
, and the homomorphisms ''f''
* are bijective for all ''x'' in ''X'' and all ''n'' ≥ 1. (For ''X'' and ''Y''
path-connected, 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 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, namely the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
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, 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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstr ...
, 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