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| In [[mathematics]], in the area of [[algebraic topology]], '''simplicial homology''' is a theory with a [[finitary]] definition, and is probably the most tangible variant of [[homology theory]].
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| Simplicial homology concerns [[topological spaces]] whose building blocks are ''n''-[[simplex]]es, the ''n''-dimensional analogs of triangles. By definition, such a space is [[homeomorphic]] to a [[simplicial complex]] (more precisely, the [[geometric realization]] of an [[abstract simplicial complex]]). Such a homeomorphism is referred to as a ''[[Triangulation (topology)|triangulation]]'' of the given space. Replacing ''n''-simplexes by their continuous images in a given topological space gives [[singular homology]]. The simplicial homology of a simplicial complex is naturally isomorphic to the [[singular homology]] of its geometric realization. This implies, in particular, that the simplicial homology of a space does not depend on the triangulation chosen for the space.
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| By for instance, [[Morse theory]], it can be seen that all smooth [[manifold]]s allow a triangulation. This, together with the fact that it is possible to resolve the simplicial homology of a simplicial complex automatically and efficiently, make this theory feasible for application to real life situations, such as [[image analysis]], [[medical imaging]], and [[data analysis]] in general.
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| == Definition ==
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| Let ''S'' be a simplicial complex. A [[Chain (algebraic topology)|simplicial ''k''-chain]] is a [[free abelian group#formal sum|formal sum]] of ''k''-simplices | |
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| :<math>\sum_{i=1}^N c_i \sigma^i \,</math>, where <math>c_i \in \mathbb{Z}, \sigma^i \in S</math> is the ''i''-th ''k''-simplex.
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| The group of ''k''-chains on ''S'', the [[free abelian group]] with basis the set of ''k''-simplices in ''S'', is denoted ''C<sub>k</sub>''.
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| Consider that a basis element of ''C<sub>k</sub>'', a ''k''-simplex, is given by a [[tuple]] of 0-simplices, or vertices
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| : <math>\sigma = \left \langle v^0 , v^1 , \dots ,v^k\right \rangle.</math>
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| The boundary operator
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| :<math>\partial_k: C_k \rightarrow C_{k-1}</math>
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| is a homomorphism defined by:
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| :<math>\partial_k(\sigma)=\sum_{i=0}^k (-1)^i \left \langle v^0 , \dots , \widehat{v^i} , \dots ,v^k\right \rangle ,</math>
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| where the simplex
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| :<math>\left \langle v^0 , \dots , \widehat{v^i} , \dots ,v^k\right \rangle</math>
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| is the ''i''<sup>th</sup> face of ''σ'' obtained by deleting its ''i''<sup>th</sup> vertex. | |
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| In ''C<sub>k</sub>'', elements of the subgroup
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| :<math>Z_k = \ker \partial_k</math>
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| are referred to as '''cycles''', and the subgroup
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| :<math>B_k = \operatorname{im} \partial_{k+1}</math>
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| is said to consist of '''boundaries'''.
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| Direct computation shows that ''B<sub>k</sub>'' lies in ''Z<sub>k</sub>'', that is, ''B<sub>k</sub>'' ⊆ ''Z<sub>k</sub>''. The boundary of a boundary must be zero. In other words,
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| :<math>(C_k, \partial_k)</math>
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| form a simplicial [[chain complex]].
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| [[Image:Triangles for simplical homology.jpg|thumb|100 px| A simplicial complex with 2 1-holes]]
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| The ''k''<sup>th</sup> homology group ''H<sub>k</sub>'' of ''S'' is defined to be the [[quotient group|quotient]]
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| :<math>H_k(S) = Z_k/B_k\, .</math>
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| A homology group ''H<sub>k</sub>'' is not trivial if the complex at hand contains ''k''-cycles which are not boundaries. This indicates that there are ''k''-dimensional holes in the complex. For example consider the complex obtained by gluing two triangles (with no interior) along one edge, shown in the image. This is a triangulation of the figure eight. The edges of each triangle form a cycle. These two cycles are by construction not boundaries (there are no 2-chains). Therefore the figure has two "1-holes".
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| Holes can be of different dimensions. The [[rank of an abelian group|rank]] of the homology groups, the numbers
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| :<math>\beta_k = {\rm rank} (H_k(S))\,</math>
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| are referred to as the [[Betti numbers]] of the space ''S'', and gives a measure of the number of ''k''-dimensional holes in ''S''.
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| ===Example===
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| [[Image:SimplexTriangle.png|thumb|150 px| The 2D complex - the triangle]]
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| In order to compute the homology groups of the triangle, one should compute the different groups <math>\mathrm{ker}(\partial_0),\mathrm{Im}(\partial_1)</math> etc. Here, by the definition of the boundary operator, we have <math>\partial_0([v_i]) = 0 </math>, therefore the kernel is:
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| :<math>\mathrm{ker}(\partial_0) = C_0 = \{a_1[v_1] + a_2[v_2] + a_3[v_3] | a_1,a_2,a_3 \in \mathbb{Z}\} \cong \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}</math>
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| that is every 0-chain is in the kernel. Next, given a 1-chain <math>c_1 = b_1[v_1,v_2] + b_2[v_2,v_3] + b_3[v_3,v_1]</math> there exists:
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| :<math>\partial_1(c_1) = (b_3-b_1)[v_1] + (b_1-b_2)[v_2] + (b_2-b_3)[v_3]</math>
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| That is,
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| :<math>\mathrm{Im}(\partial_1) = \{(b_3-b_1)[v_1] + (b_1-b_2)[v_2] + (b_2-b_3)[v_3] | b_1,b_2,b_3\in \mathbb{Z}\}</math>,
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| which means that a 0-chain <math>c_0 = a_1[v_1] + a_2[v_2] + a_3[v_3]</math> is in the image of <math>\partial_1</math> if and only if
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| :<math>a_1 = b_3-b_1</math>
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| :<math>a_2 = b_1-b_2</math>
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| :<math>a_3 = b_2-b_3</math>.
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| This implies that we have only two degrees of freedom for choosing <math>a_i</math>, or in other words:
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| :<math>\mathrm{Im}(\partial_1) \cong \mathbb{Z} \oplus \mathbb{Z}</math>
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| Now we can use the definition:
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| :<math>H_0(S) \cong (\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z})/ (\mathbb{Z} \oplus \mathbb{Z}) \cong \mathbb{Z}</math>
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| As for the other homology groups, computations are easier. <math>\partial_1(c_1) = 0</math> if and only if <math>b_1=b_2=b_3</math>, therefore
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| :<math>\mathrm{ker}(\partial_1) = \{b[v_1,v_2] + b[v_2,v_3] + b[v_3,v_1] | b\in \mathbb{Z}\} \cong \mathbb{Z}.</math>
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| Now, since there are no 2-chains, the kernel and image of <math>\partial_2</math> are trivial, that is <math>\mathrm{ker}(\partial_2) = \mathrm{Im}(\partial_2) = 0</math>. This yields:
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| :<math> H_1(S) = \mathrm{ker}(\partial_1) / \mathrm{Im}(\partial_2) = \mathrm{ker}(\partial_1) \cong \mathbb{Z}</math>
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| :<math> H_2(S) = \mathrm{ker}(\partial_2) / \mathrm{Im}(\partial_3) \cong 0</math>
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| == Applications ==
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| A standard scenario in many computer applications is a collection of points (measurements, dark pixels in a bit map, etc.) in which one wishes to find a topological feature. Homology can serve as a qualitative tool to search for such a feature, since it is readily computable from combinatorial data such as a simplicial complex. However, the data points have to first be [[Triangulation_(topology)|triangulated]], meaning one replaces the data with a simplicial complex approximation. Computation of [[persistent homology]] ([http://graphics.stanford.edu/projects/lgl/paper.php?id=elz-tps-02 Edelsbrunner et al.2002 ][http://at.yorku.ca/b/a/a/k/28.htm Robins, 1999]) involves analysis of homology at different resolutions, registering homology classes (holes) that persist as the resolution is changed. Such features can be used to detect structures of molecules, tumors in X-rays, and cluster structures in complex data. A [[MATLAB]] toolbox for computing persistent homology, Plex ([[Vin de Silva]], [[Gunnar Carlsson]]), is available at [http://math.stanford.edu/comptop/programs/ this site] Stand-alone implementations in [[C++]] are available as part of the [http://www.sas.upenn.edu/~vnanda/perseus/index.html Perseus] and [http://www.mrzv.org/software/dionysus/ Dionysus] software projects. More generally, simplicial homology plays a central role in [[topological data analysis]], a technique in the field of [[data mining]].
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| ==See also==
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| *[[Homology theory]]
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| *[[Singular homology]]
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| *[[Cellular homology]]
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| == References ==
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| *Lee, J.M., ''Introduction to Topological Manifolds'', [[Springer-Verlag]], Graduate Texts in Mathematics, Vol. 202 (2000) ISBN 0-387-98759-2
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| *[[Allen Hatcher|Hatcher, A.]], ''[http://www.math.cornell.edu/~hatcher/AT/ATchapters.html Algebraic Topology],'' [[Cambridge University Press]] (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
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| *Moise, E.E., ''Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung.'' Ann. Math. 96-114 (1952).
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| ==External links==
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| *[http://math.stanford.edu/comptop/ Topological methods in scientific computing]
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| *[http://www.math.gatech.edu/~chomp/ Computational homology (also cubical homology)]
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| [[Category:Homology theory]]
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| [[Category:Algebraic topology]]
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| [[Category:Computational topology]]
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