Editing Homology (mathematics) (section)

== Applications ==

=== Application in pure mathematics ===
Notable theorems proved using homology include the following:
* The [[Brouwer fixed point theorem]]: If ''f'' is any continuous map from the ball ''B<sup>n</sup>'' to itself, then there is a fixed point <math>a \in B^n</math> with <math>f(a) = a.</math>
* [[Invariance of domain]]: If ''U'' is an [[open set|open subset]] of <math>\R^n</math> and <math>f : U \to \R^n</math> is an [[injective]] [[continuous map]], then <math>V = f(U)</math> is open and ''f'' is a [[homeomorphism]] between ''U'' and ''V''.
* The [[Hairy ball theorem]]: any continuous vector field on the 2-sphere (or more generally, the 2''k''-sphere for any <math>k \geq 1</math>) vanishes at some point.
* The [[Borsuk–Ulam theorem]]: any [[continuous function]] from an [[n-sphere|''n''-sphere]] into [[Euclidean space|Euclidean ''n''-space]] maps some pair of [[antipodal point]]s to the same point. (Two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.)
* Invariance of dimension: if non-empty open subsets <math>U \subseteq \R^m</math> and <math>V \subseteq \R^n</math> are homeomorphic, then <math>m = n.</math>{{sfn|Hatcher|2002|p=126}}

=== Application in science and engineering ===

In [[topological data analysis]], data sets are regarded as a [[point cloud]] sampling of a manifold or [[algebraic variety]] embedded in [[Euclidean space]]. By linking nearest neighbor points in the cloud into a triangulation, a simplicial approximation of the manifold is created and its simplicial homology may be calculated. Finding techniques to robustly calculate homology using various triangulation strategies over multiple length scales is the topic of [[persistent homology]].<ref name=CompTop>{{cite web|title=CompTop overview|url=http://comptop.stanford.edu/|access-date=16 March 2014|archive-date=22 June 2007|archive-url=https://web.archive.org/web/20070622154434/http://comptop.stanford.edu/|url-status=dead}}</ref>

In [[sensor network]]s, sensors may communicate information via an ad-hoc network that dynamically changes in time. To understand the global context of this set of local measurements and communication paths, it is useful to compute the homology of the [[network topology]] to evaluate, for instance, holes in coverage.<ref>{{cite web|title=Robert Ghrist: applied topology|url=http://www.math.upenn.edu/~ghrist/research.html|access-date=16 March 2014}}</ref>

In [[dynamical system]]s theory in [[physics]], Poincaré was one of the first to consider the interplay between the [[invariant manifold]] of a dynamical system and its topological invariants. [[Morse theory]] relates the dynamics of a gradient flow on a manifold to, for example, its homology. [[Floer homology]] extended this to infinite-dimensional manifolds. The [[KAM theorem]] established that [[periodic orbit]]s can follow complex trajectories; in particular, they may form [[Braid theory|braids]] that can be investigated using Floer homology.<ref name=vandenBerg2014>
{{cite journal
 |last1 = van den Berg |first1 = J.B.
 |last2 = Ghrist |first2 = R.
 |last3 = Vandervorst |first3 = R.C.
 |last4 = Wójcik |first4 = W.
 |s2cid = 16865053
 |title=Braid Floer homology
 |journal=Journal of Differential Equations
 |date=2015
 |volume=259
 |issue=5
 |pages=1663–1721
 |doi=10.1016/j.jde.2015.03.022
 |bibcode = 2015JDE...259.1663V
 |url=http://www.math.vu.nl/~janbouwe/pub/braidfloerhomology.pdf
 |doi-access = free
}}</ref>

In one class of [[finite element methods]], [[boundary-value problem]]s for differential equations involving the [[Hodge-Laplace operator]] may need to be solved on topologically nontrivial domains, for example, in [[Computational electromagnetics|electromagnetic simulation]]s. In these simulations, solution is aided by fixing the [[cohomology class]] of the solution based on the chosen boundary conditions and the homology of the domain. FEM domains can be triangulated, from which the simplicial homology can be calculated.<ref name=Pellikka2013>{{cite journal|last=Pellikka|first=M|author2=S. Suuriniemi |author3=L. Kettunen |author4=C. Geuzaine |title=Homology and Cohomology Computation in Finite Element Modeling|journal=SIAM J. Sci. Comput.|date=2013|volume=35|issue=5|pages=B1195–B1214|doi=10.1137/130906556|bibcode=2013SJSC...35B1195P|citeseerx=10.1.1.716.3210|url=http://geuz.org/gmsh/doc/preprints/gmsh_homology_preprint.pdf}}</ref><ref name=Arnold2006>{{cite journal|last=Arnold|first=Douglas N. |author2=Richard S. Falk |author3=Ragnar Winther|title=Finite element exterior calculus, homological techniques, and applications|journal=Acta Numerica|date=16 May 2006|volume=15|pages=1–155|doi=10.1017/S0962492906210018|bibcode=2006AcNum..15....1A |s2cid=122763537 |url=http://purl.umn.edu/4216 }}</ref>