In
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 ...
, the cellular approximation theorem states that a
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
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 clas ...
es can always be taken to be of a specific type. Concretely, if ''X'' and ''Y'' are CW-complexes, and ''f'' : ''X'' → ''Y'' is a continuous map, then ''f'' is said to be ''cellular'', if ''f'' takes the
''n''-skeleton of ''X'' to the ''n''-skeleton of ''Y'' for all ''n'', i.e. if
for all ''n''. The content of the cellular approximation theorem is then that any continuous map ''f'' : ''X'' → ''Y'' between CW-complexes ''X'' and ''Y'' is
homotopic
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 deforma ...
to a cellular map, and if ''f'' is already cellular on a subcomplex ''A'' of ''X'', then we can furthermore choose the homotopy to be stationary on ''A''. From an algebraic topological viewpoint, any map between CW-complexes can thus be taken to be cellular.
Idea of proof
The proof can be given by
induction after ''n'', with the statement that ''f'' is cellular on the skeleton ''X''
''n''. For the base case n=0, notice that every
path-component of ''Y'' must contain a 0-cell. The
image under ''f'' of a 0-cell of ''X'' can thus be connected to a 0-cell of ''Y'' by a path, but this gives a homotopy from ''f'' to a map which is cellular on the 0-skeleton of X.
Assume inductively that ''f'' is cellular on the (''n'' − 1)-skeleton of ''X'', and let ''e''
''n'' be an ''n''-cell of ''X''. The
closure of ''e''
''n'' is
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 ...
in ''X'', being the image of the characteristic map of the cell, and hence the image of the closure of ''e''
''n'' under ''f'' is also compact in ''Y''. Then it is a general result of CW-complexes that any compact subspace of a CW-complex meets (that is,
intersects non-trivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a ...
ly) only finitely many cells of the complex. Thus ''f''(''e''
''n'') meets at most finitely many cells of ''Y'', so we can take
to be a cell of highest dimension meeting ''f''(''e''
''n''). If
, the map ''f'' is already cellular on ''e''
''n'', since in this case only cells of the ''n''-skeleton of ''Y'' meets ''f''(''e''
''n''), so we may assume that ''k'' > ''n''. It is then a technical, non-trivial result (see Hatcher) that the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
of ''f'' to
can be
homotoped relative to ''X''
''n-1'' to a map missing a point ''p'' ∈ ''e''
''k''. Since ''Y''
''k'' −
deformation retracts onto the subspace ''Y''
''k''-''e''
''k'', we can further homotope the restriction of ''f'' to
to a map, say, ''g'', with the property that ''g''(''e''
''n'') misses the cell ''e''
''k'' of ''Y'', still relative to ''X''
''n-1''. Since ''f''(''e''
''n'') met only finitely many cells of ''Y'' to begin with, we can repeat this process finitely many times to make
miss all cells of ''Y'' of dimension larger than ''n''.
We repeat this process for every ''n''-cell of ''X'', fixing cells of the subcomplex ''A'' on which ''f'' is already cellular, and we thus obtain a homotopy (relative to the (''n'' − 1)-skeleton of ''X'' and the ''n''-cells of ''A'') of the restriction of ''f'' to ''X''
''n'' to a map cellular on all cells of ''X'' of dimension at most ''n''. Using then the
homotopy extension property to extend this to a homotopy on all of ''X'', and patching these homotopies together, will finish the proof. For details, consult Hatcher.
Applications
Some homotopy groups
The cellular approximation theorem can be used to immediately calculate some
homotopy groups. In particular, if