A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology.[1] It was introduced by J. H. C. Whitehead[2] to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.[2]
^Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. ISBN 0-521-79540-0. This textbook defines CW complexes in the first chapter and uses them throughout; includes an appendix on the topology of CW complexes. A free electronic version is available on the author's homepage.
^ abWhitehead, J. H. C. (1949a). "Combinatorial homotopy. I." (PDF). Bulletin of the American Mathematical Society. 55 (5): 213–245. doi:10.1090/S0002-9904-1949-09175-9. MR 0030759. (open access)
A CWcomplex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was...
Look up CW in Wiktionary, the free dictionary. CW may stand for: centiwatt (cW), one hundredth of a watt Cω, a programming language CWcomplex, a type...
minimal regular CW structure on the sphere. In light of the smooth structure, the existence of a Morse function would show RPn is a CWcomplex. One such function...
compression, and topological data analysis. Let X {\displaystyle X} be a CWcomplex and denote by X {\displaystyle {\mathcal {X}}} its set of cells. Define...
equivalence from a CWcomplex, this axiom reduces homology or cohomology theories on all spaces to the corresponding theory on CWcomplexes. Some examples...
Simplicial complex, a kind of topological space CWcomplex, a kind of topological space Line complex, a 3-dimensional family of lines in space Complex (geology)...
line with a single point removed. Real projective spaces have a simple CWcomplex structure, as Pn(R) can be obtained from Pn−1(R) by attaching an n-cell...
analytic varieties, semialgebraic sets, and subanalytic sets. CW-complexes A CWcomplex is a topological space formed by gluing disks of different dimensionality...
spaces. Algebraic topologists work with compactly generated spaces, CWcomplexes, or spectra. Formally, a homotopy between two continuous functions f...
the Whitehead theorem states that if a continuous mapping f between CWcomplexes X and Y induces isomorphisms on all homotopy groups, then f is a homotopy...
of the classifying space takes place within the homotopy category of CWcomplexes, existence theorems for universal bundles arise from Brown's representability...
states that a map between CW-complexes 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...
of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. If X {\displaystyle X} is a CW-complex...
and is called the Bruschlinsky group. Provided X {\displaystyle X} is a CW-complex, it is isomorphic to the first cohomology group H 1 ( X ) {\displaystyle...
extra constraints, such as being compactly generated, or Hausdorff, or a CWcomplex. In the same vein as above, a "map" is a continuous function, possibly...
purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CWcomplex is a type of topological space introduced by J...
finite CW-complexes. (When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the...
cellular space is a compact Hausdorff space that has the structure of a CWcomplex. "Naturally reductive homogeneous spaces and homogeneous structures of...
to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though...
slightly. For example, the smash product of two CWcomplexes is a CWcomplex if one uses the product of CWcomplexes in the definition rather than the product...
CWcomplexes in homotopy theory are generalized by analogous results for simplicial sets. While algebraic topologists largely continue to prefer CW complexes...
{CP} ^{\infty }]} for any nice CW-complex X {\displaystyle X} . Moreover, from the theory of Chern classes, every complex line bundle L → X {\displaystyle...
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