# Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive arrows is zero seems like a fairly general notion, but I have not come across it in fields like biology, economics, etc. Are there examples of non-trivial (co)homology appearing outside of pure mathematics?

I think Hatcher has a couple illustrations of homology in his textbook involving electric circuits. This is the type of thing I'm looking for, but it still feels like topology since it is about closed loops. Since the relation $d^2=0$ seems so simple to state, I would imagine this setup to be ubiquitous. Is it? And if not, why is it so special to topology and related fields?

• Ghrist has written a number of papers applying homology to different applied fields. See for example his paper on Sensor Networks. – Jim Conant Mar 30 '11 at 19:30
• This question should be Community Wiki, since it's looking for a list of examples rather than an answer to a specific question. I also feel like the question isn't very clear, since any answer has to be homological in nature so fundamentally mathematical. So the question seems somewhat conflicted. – Ryan Budney Mar 30 '11 at 19:35
• Do you count string theory as "outside the realm of pure mathematics"? – Qfwfq Mar 30 '11 at 20:04
• The first paragraph seems to suggest that the question is not so much about homology/topology in the large, but more about the idea of chain complexes. Is my reading correct? – Yemon Choi Mar 30 '11 at 20:27
• Applied topology at Stanford: comptop.stanford.edu – Igor Belegradek Mar 30 '11 at 20:48

Robert Ghrist is all about applied topology: Sensor Network, Signal Processing, and Fluid Dynamics. (homepage: http://www.math.upenn.edu/~ghrist/index.html ). For instance, we want to use the least number of sensors to cover a certain area, such that when we remove one sensor, a part of that area is undetectable. We can form a complex of these sensors and hence its nerve, and use homology to determine whether there are any gaps in the sensor-collection. I've met with him in person and he expressed confidence that this is going to be a big thing of the future.

There are also applications of cohomology to Crystallography (see Howard Hiller) and Quasicrystals in physics (see Benji Fisher and David Rabson). In particular, it uses cohomology in connection with Fourier space to reformulate the language of quasicrystals/physics in terms of cohomology... Extinctions in x-ray diffraction patterns and degeneracy of electronic levels are interpreted as physical manifestations of nonzero homology classes.

Another application is on fermion lattices (http://arxiv.org/abs/0804.0174v2), using homology combinatorially. We want to see how fermions can align themselves in a lattice, noting that by the Pauli Exclusion principle we cannot put a bunch of fermions next to each other. Homology is defined on the patterns of fermion-distributions.

• Robert Ghrist is coming to Edinburgh for the Science Festival this year to talk about the Mathematics of holes. (Alas I will be away.) If anyone is around, it will be worth your while to attend! – José Figueroa-O'Farrill Mar 30 '11 at 20:59
• In general, I think that homology will play a role in the mathematics of information. We had a talk by Gunnar Carlsson recently in Edinburgh about "persistent homology" and it was quite an eye opener. See comptop.stanford.edu for instance. – José Figueroa-O'Farrill Mar 30 '11 at 21:00

Actually even schoolchildren calculate group co-cycle. (Without knowing that it is called like this). Cohomology occurs in everyday life as soon as one learns to count.

5+7 = 1 2

4 + 5 = 0 9

2 + 8 = 1 0

What is the function on which sends a pair (a,b) to the $0$ or $1$ depending result is greater than 9 or not ? ( e.g. f(5,7)= 1, f(4,5) = 0, f(2,8)= 1).

This is actually a 2-cocycle for group $Z/nZ$ with values in $Z$.

It can be checked directly or...

Let us look on it more conceptually. Consider the standard short exact sequence of abelian groups $0 \to Z \to Z \to Z/n \to 0$. (First map is multiplication by $n$, the second is factorization and will be denoted by $p$).

Choose section $s: Z/nZ \to Z$ (i.e. any map such $ps=Id$, where $p: Z \to Z/nZ$, it is like connection in differential geometry (can be made precise)).

Define $f(a,b)= s(a)+s(b) - s(a+b)$

Note that: a) this function $f(a,b)$ is exactly we talked above

b) from general theory this is 2-cocyle, (it corresponds to this extension, (it it is like "curvature" of connection is differential geomety (can be made precise)).

That is all: we explained why it is group cocycle and what its role.

I would like to learn this 20 years ago when I learned group cohomology as undergraduate, but I learned this 1 ago, doing some engineering work in wireless communication... I am still surprised that it is not written on the first page of any textbook which deals with group cohomology, when I am explaining this to my friends most did not know this also and after knowing share my feeling of surprise.

• While not on the first page of a textbook, this is written up in an article in the American Mathematical Monthly (Daniel C. Isaksen, A cohomological viewpoint on elementary school arithmetic, Amer. Math. Monthly (109), no. 9 (2002), p. 796--805) – Christopher Drupieski Jun 11 '12 at 20:44
• is it easy to generalize to larger digits? how about for multiplication? – Turbo Jun 18 '12 at 16:34
• @36min In arxiv.org/abs/hep-th/0212195 we have discussed such a look on connections (actually for more general setup of Courant algebroids), it might be possible to get idea from there, but may be it is not good starting point. I am sorry I do not know reference for standard exposition of this approach in the case of usual connections. – Alexander Chervov Nov 21 '12 at 6:16
• @36min Let me give an idea consider module V over M. Consider the following exact triple: End(V) ->A(V) -> Der(M), where A(M) - the set of all "derivations" of module V. The idea is that connection is exactly the same is a section s: A(V) <- Der(M) !!! The curvature is the following - take two elements a,b in Der(M) , consider F(a,b) = [s(a),s(b)] -s([a,b]) observe that F(a,b) lies actually in End(V) \subset A(V) so we get map F(a,b) : Der(M)^2 -> End(V) - this is curvature – Alexander Chervov Nov 21 '12 at 6:24
• @36min , what is Der(M) - set of all derivations of algebra of functions on M, i.e. all vector fields. What is A(V) - set of all derivations of module V. Map d: V-> V is called derivation of module V, if there exists derivation dd \in Der(M) such that for any element "a" of algebra M , it is true that d(a v) = dd(a) v + a dd(v) !!!!!!!! It is simple, may be I am explaining not in a right way. Is it clear ? – Alexander Chervov Nov 21 '12 at 6:28

Recently, it was realized that quantum many-body states can be divided into short-range entangled states and long-range entangled states.

The quantum phases with long-range entanglements correspond to topologically ordered phases, which, in two spatial dimensions, can be described by tensor category theory (see cond-mat/0404617). Topological order in higher dimensions may need higher category to describe them.

One can also show that the quantum phases with short-range entanglements and symmetry $G$ in any dimensions can be "classified" by Borel group cohomology theory of the symmetry group. (Those phases are called symmetry protected trivial (SPT) phases.)

The quantum phases with short-range entanglements that break the symmetry are the familar Landau symmetry breaking states, which can be described by group theory.

So, to understand the symmetry breaking states, physicists have been forced to learn group theory. It looks like to understand patterns of many-body entanglements that correspond to topological order and SPT order, physicists will be forced to learn tensor category theory and group cohomology theory. In modern quantum many-body physics and in modern condensed matter physics, tensor category theory and group cohomology theory will be as useful as group theory. The days when physics students need to learn tensor category theory and group cohomology theory are coming, may be soon.

Quantum field theory is outside the realm of pure mathematics, makes contact with the real world and features chain complexes and cohomology.

The current paradigm for gauge theories such as the standard model is based on Yang-Mills theories coupled to matter. The quantisation of nonabelian (and, depending on your choice of gauge fixing function, also abelian) Yang-Mills theories features a cohomology theory known by the moniker of BRST, after the inventors: Becchi, Rouet, Stora and, independently, Tyutin. The cleanest proofs of the renormalizability of Yang-Mills theories are cohomological in nature.

My understanding, from conversations with Raoul Bott, is that his early work on electrical circuits and the Bott-Duffin theorem can be intepreted as exhibiting close connections between de Rham cohomology and the laws of electrical circuits, and that this is part of what led him into pure mathematics early in his career.

• He talked about this connection to electrical circuits in teaching graduate algebraic topology, to help motivate cohomology and give intuition for it. – Patricia Hersh May 20 '12 at 14:25
• Bott gave a very nice talk (c. 1960) to electrical engineers on the subject, but I cannot find a reference right now. – Robert Bruner Jun 11 '12 at 21:55
• See the paper Geometry and Topology: Seven Lectures by Raoul Bott edited by J. M. Rojas. – Tom Copeland Mar 30 '14 at 23:13

The mass of a classical mechanical system is an element in the (one-dimensional) second cohomology group of the Lie algebra of the Galilei group. See J. M. Souriau, Stucture des Systèmes Dynamiques, Chap. III, section (12.136). Or in english translation, search inside here for "total mass".

• A layman question. You say second cohomology group, however Souriau does not tell a number for it. Similarly to you, Guillemin and Sternberg tells "second" ($H^2$)(Symplectic Techniques in Physics, p. 417), but Liberman and Marle tells first ($H^1$) (Symplectic Geometry and Analytical Mechanics, p.204). Could you resolve this apparent contradiction in a couple of words? – mma Nov 24 '16 at 6:13
• As @mma points out, the note on p. 107 explicitly mentions that Souriau's "cohomology" is always "cohomology in degree 1". (I can't manage to trace through his definitions to tell in what sense $m$ is a cocycle at all, regardless of degree.) – LSpice Jun 10 '17 at 20:28

The Aharonov–Bohm effect. Classically, you can't distinguish two electromagnetic potentials which are in the same cohomology class. From quantum viewpoint, they can be distinguished, because an electron changes its phase under parallel transport defined by the connection associated to a potential.

The finite element method- a numerical method for solving PDE's- has a homological interpretation:

MR2269741 (2007j:58002) Arnold, Douglas N.(1-MN-MA); Falk, Richard S.(1-RTG); Winther, Ragnar(N-OSLO-CMA) Finite element exterior calculus, homological techniques, and applications. (English summary) Acta Numer. 15 (2006), 1–155

1. Something resembling de Rham complex with differential-algebraic flavor appears in (variant of) control theory, see, for example, G. Conte, C.H. Moog, A.M. Perdon, Algebraic Methods for Nonlinear Control Systems, 2nd ed., Springer, 2006. But, as far as I can tell, they do not use the word "cohomology" explicitly.

2. Spencer cohomology (which is, essentially, a Lie algebra cohomology) appears as obstructions to integrability of some differential-geometric structures (G-structures) and, through it, of (some) differential equations. Potentially this opens a wide possibilities for applications, and indeed, Dimitry Leites advocates this approach in (some of) his writings. An emblematic publication which is available, unfortunately, only in Russian, is: "Application of cohomology of Lie algebras in national economy", Seminar "Globus", Independent Univ. of Moscow, Vol. 2, 2005, 82-102. The Russian original for "national economy" in the title is (a somewhat pejorative and untranslatable term) "narodnoe khozyai'stvo".

Edit: J.-F. Pommaret has published extensively on the applications of Spencer cohomology to continuum mechanics, control theory and mathematical physics - for example, see "Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics"

• I love that there be a pejorative form of "national economy" ;-) – Mariano Suárez-Álvarez Apr 20 '14 at 2:04

A classical and elegant application is to the solution of Kirchhoff's theorem on electrical cricuits. See:

Nerode, A.; Shank, H.: An algebraic proof of Kirchhoff's network theorem. Amer. Math. Monthly, 68 (1961) 244–247

• This is discussed in more detail in Quantum Field Theory III: Gauge Theory by Eberhard Zeidler (Springer, 4/2011), pg. 1009, 1020, 1027. – Tom Copeland Jan 24 '17 at 18:56
• @TomCopeland The source you gave works with both chain and cochain complexes at the same time. It seems to me that it's merely reproducing the Bott-Duffin approach. The advantage of the Nerode-Shank argument is that it easily generalizes to higher dimensions. See: Catanzaro, Michael J.(1-WYNS); Chernyak, Vladimir Y.(1-WYNS-KM); Klein, John R.(1-WYNS) Kirchhoff's theorems in higher dimensions and Reidemeister torsion. Homology Homotopy Appl. 17 (2015), no. 1, 165–189. – John Klein Jan 25 '17 at 20:48

It's my understanding that Carina Curto and Vladmir Itskov at the University of Nebraska - Lincoln apply algebraic topology (among other things) to study theoretical and applied neuroscience.

• Their work is very similar in spirit to Ghrist's work alluded to in one of the other answers. – Igor Rivin May 20 '12 at 15:38

Anders Björner and László Lovász used bounds on the Betti numbers for the complement of a real subspace arrangement called the $k$-equal arrangement to give a complexity theory lower bound that agreed, up to a scalar multiple, with the previously known upper bound in:

A. Björner and L. Lovász, Linear decision trees, subspace arrangements, and Mobius functions, Journal of the American Mathematical Society, Vol. 7, No. 3 (1994), 677--706.

The basic question addressed in their paper (along with other questions of a similar flavor) is how many pairwise comparisons of coordinates are needed to decide if a vector in ${\bf R}^n$ has $k$ coordinates all equal to each other for fixed $k$ and $n$. They observed that this is equivalent to deciding whether the vector lies on the so-called $k$-equal arrangement or in its complement, where the $k$-equal arrangement is the subspace arrangement comprised of the ${n\choose k}$ subspaces where $k$ coordinates are set equal to each other.

To this end, they gave a lower bound on the number of leaves in a linear decision tree -- a tree where one starts at the root, and each time one does a comparison of two coordinates $a_i$ and $a_j$, then one proceeds down to either the $a_i < a_j$ child or the $a_i = a_j$ child or the $a_i > a_j$ child. One reaches a leaf when no further queries are necessary to make a decision as to containment in the arrangement or its complement. The log base 3 of the number of leaves is a lower bound on the depth of the tree, i.e. on the number of queries needed in the worst case.

To get some intuition for why this bound depended fundamentally on the Betti numbers of the complement, consider the $k=2$ case -- where the number of connected components of the complement of the subspace arrangement (which in this case is a hyperplane arrangement) is an obvious lower bound on the number of leaves in any linear decision tree.

Maurice Herlihy and Nir Shavit won the 2004 Gödel Prize for topological analysis of asynchronous computation. Homology was involved.

Cohomology has appeared in game theoretic research on equilibrium refinements. Loosely speaking, John Nash's 1951 original notion of an equilibrium point does too little to limit the set of 'reasonable' outcomes. A variety of so-called 'equilibrium refinements' sprang up in the economic literature intended to address this. A common theme in many of them was robustness to perturbation (if one perturbs the underlying game played, one would wish that 'nearby fixed point problems have nearby solutions').

In 1989 JF Mertens formulated a notion of a stable equilibrium over two papers that relies on the cohomological essentiality of a projection map from the graph of the equilibrium correspondence to the space of games (the two papers may be found here and here) to arrive at a solution concept with a number of normatively reasonable properties.

To add a touch of very belated whimsy, cohomology has manifestations in art. Here are two:

1. A Penrose triangle (an "impossible figure") can be viewed as a Čech $1$-cycle associated to an open covering of the image plane, taking values in the sheaf of positive functions under pointwise multiplication.

In more detail, place your eye at the origin of Cartesian space, let $P$ be a region in some plane (the screen) not through the origin, and let $C \simeq P \times (0, \infty)$ (for cone) be the union of open rays from your eye to a point of $P$.

A spatial scene in $C$ is rendered in the plane region $P$ by radial projection, discarding depth (distance) information. Recovering a spatial scene from its rendering on paper amounts to assigning a depth to each point of $P$, i.e., to choosing a section of the projection $C \to P$.

Roger Penrose's three-page article On the Cohomology of Impossible Figures describes in detail (copiously illustrated with Penrose's inimitable drawings) how a two-dimensional picture can be "locally consistent" (each sufficiently small piece is the projection of a visually-plausible spatial object) yet "globally inconsistent" (the entire picture has no visually-plausible interpretation as a projection of a spatial object).

For example, each corner of a Penrose triangle has a standard interpretation as a plane projection of an L-shaped object with square cross-section, but the resulting local depth data cannot be merged consistently in a way conforming to visual expectations.

Of course, the depth data can be merged consistently (from a carefully-selected origin!) by circumventing visual expectations. There are at least two "natural" approaches: use curved sides, or break one vertex.

2. M. C. Escher's Print Gallery is a planar, "self-including" rendition of the complex exponential map, viewed as an infinite cyclic covering of the punctured plane. (Contrary to mathematical convention, traveling clockwise around the center in Escher's print "goes up one level".) I know of no better reference that Hendrik Lenstra's and Bart de Smit's 2003 analysis of the Droste effect.

It mentions stuff like Geometric Complexity Theory, a far-out program for proving P!=NP with algebraic geometry.

Mentions the thing I actually first websearched for, Herlihy's work on concurrent and distributed computing using cohomology.

There is also an argument by Roman Jackiw that 3-cocycles appear in the quantum mechanics of a charged particle in the field of a magnetic monopole. There it happens that you can generate spatial translations either with the canonical momentum, which is gauge dependent because it involves the vector potential, or with the velocity, which is gauge invariant, because both have the same commutation relations with position. If you want to generate translations in a gauge invariant way, then translations turn out to be non-associative, i.e., you can compose 3 translation operators in different orders and get different results, all differing by a 3-cocycle in a particular cohomology. Restoring associativity leads to Dirac's quantization condition. This condition can also be obtained by different cohomological arguments, as Orlando Alvarez does, considering certain ambiguities that appear when you integrate a connection 1-form along paths covered by two or three patches. It happens that it is not trivial how to do this, you cannot just integrate along one patch up to a point in the intersection and keep integrating along the other patch. You have to do extra tricks that lead to ambiguities, which can be resolved if you impose some condition which is equivalent to Dirac's quantization condition when the connection is the vector potential of a monopole. The condition that you have to impose can be recast in terms of Cech cohomology. The papers: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.54.159, http://link.springer.com/article/10.1007%2FBF01212452

The group cohomology can be useful to describe the many-body quantum Condensed Matter systems with emergent underlying intrinsic Topological Orders. These classes of systems at low energy and long distance behave like certain TQFT theories. In particular, one can write explicit exact solvable Lattice quantum Hamiltonian operator $\hat{H}$ on the space discretized lattice, and its ground states give rise to discrete gauge theory TQFT with finite group $G$ (quantum double models in 2+1d or its 3+1d, etc analogous, $D(G)$, with Drinfield and Hopf algebra), twisted discrete gauge theories (twisted quantum double models $D^\omega(G)$).

See:

The low energy and long distance physics of the theories are the same as the Dijkgraaf-Witten topological gauge theories, Kitaev toric code and quantum double models, and some of Levin-Wen string-net models, and many of twisted discrete gauge theory.

This is how the quantum Hamiltonian operator (acting on the Hilbert space of quantum states) looks like: $$\hat{H}=-\sum_v A_v-\sum_f B_f,$$ where $B_f$ is the face operator defined at each triangular face $f$, and $A_v$ is the vertex operator defined on each vertex $v$. As in the TQD model in $(2+1)$-d, each operator $A_v$ behaves as a gauge transformation on the group elements respectively on the edges meeting at $v$, and a $B_f$ detects whether the flux through face $f$ is zero. This kind of Hamiltonians generically feature ground states that are gauge invariant and bear zero flux everywhere.

A normalized $$\omega_d\in H^d(G,U(1)),$$ as a function $\omega:G^4\rightarrow U(1)$, satisfies the $d$-cocycle condition. The cocycle $\omega$ will fill into the spacetime lattice $d$-simplex ($d+1$-cell). Let us take $D=4$-dim spacetime as an example below.

The operator $A_v$ is a summation $$A_v=\frac{1}{|G|}\sum_{[vv']=g\in G}A_v^g.$$ The value $|G|$ is the order of the group $G$. The operator $A_v^g$ acts on a vertex $v$ with a group element $g\in G$ by replacing $v$ by a new enumeration $v'$ that is slightly" less than $v$ but greater than all the enumerations that are less than $v$ in the original set of enumerations before the action of the operator, such that $v'v=g$. In a dynamical picture of Hamiltonian evolution, $v'$ is understood as on the next \textquotedblleft time" slice, and there is an edge $v'v\in G$ in the $(3+1)$ dimensional \textquotedblleft spacetime" picture. That is, the new vertex $v'$ and the original vertices before the action of $A_v$ delineate a $4$-dimensional picture. Let us consider as follows the simplest subgraph---namely a single tetrahedron---of some large $\Gamma$ to illustrate how an $A_v$ acts: where $v'_4v_4=g$.

The action of $B_f$ on a basis vector is The discrete delta function $\delta_{v_1v_2\cdot v_2v_3\cdot v_3v_1}$ is unity if ${v_1v_2\cdot v_2v_3\cdot v_3v_1=1 }$, where $1$ is the identity element in $G$, and 0 otherwise. Note again that here, the ordering of $v_1,v_2$, and $v_3$ does not matter because of the identities $\delta_{v_1v_2\cdot v_2v_3\cdot v_3v_1} =\delta_{v_3v_1\cdot v_1v_2\cdot v_2v_3}$ and $\delta_{v_1v_2\cdot v_2v_3\cdot v_3v_1} =\delta_{\overline{v_1v_2\cdot v_2v_3\cdot v_3v_1}} =\delta_{\overline{v_3v_1}\cdot \overline{v_2v_3}\cdot \overline{v_1v_2}} =\delta_{v_1v_3\cdot v_3v_2\cdot v_2v_1}$. In other words, in any state on which $B_f=1$ on a triangular face $f$, the three group degrees of freedom around $v$ is related by a chain rule: $$v_1v_3=v_1v_2\cdot v_2v_3$$ for any enumeration $v_1,v_2,v_3$ of the three vertices of the face $f$.

Here are how the space lattice and the triangulation of spacetime lattices looks like:

Here are how the $SL(N,\mathbb{Z})$'s modular $S$ and $T$-transformations look like:

Jim Stasheff has surveyed the history of cohomology in physics in the 20th century: see the pdf linked at https://ncatlab.org/nlab/show/A+Survey+of+Cohomological+Physics

In (real world, for $n=3$) crystallography, given a point group $K\subseteq\mathrm{O}(n)$ and a ($K$-invariant) lattice $T\subseteq\mathbb{R}^n$,

the set of possible crystallographic classes (where, in the real world, a "crystallographic class" is unederstood as "the class of all crystals that have isomorphic isometry group"; and in general it just means a space group or crystallographic group, i.e. a discrete subgroup $G\subseteq\mathrm{Euc}(n)$ that contains $n$ linearly independent translations, up to conjugation by elements in the affine group $\mathrm{Aff(n,\mathbb{R})}$ or -which is the same by a theorem of Bieberbach- up to just abstract isomorphism)

that lie in the arithmetic crystal class determined by $(K,T)$, that is:

• have $K$ as point group (up to conjugation by $\mathrm{GL}(n,\mathbb{Z})$, where $K\subseteq\mathrm{GL}(n,\mathbb{Z})$ via a lattice basis)
• and have $T$ as translation lattice,

is in bijection with the quotient set

$$\frac{\mathrm{H}^1(K,\mathbb{R}^n/T)}{\mathcal{N}_{\mathrm{GL}(n,\mathbb{Z})}(K)}$$

where $\mathcal{N}_{\mathrm{GL}(n,\mathbb{Z})}(K)$ is the integral normalizer of $K$, acting on group cohomology $\mathrm{H}^1(K,\mathbb{R}^n/T)\simeq\mathrm{H}^2(K,T)$ (induced via the defining action of $K$ on $T$).

According to the crystallographer's terminology, crystals that correspond to the zero element in cohomology are said to be symmorphic.

In the theory of error correcting codes, linear codes are most prominent. These can be seen as the kernel of a "parity-check" matrix, most commonly working over the two element field, $F_2.$

For example, a simple three bit repetition code corresponds to the kernel of the linear operator $$\partial = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ \end{pmatrix}.$$ The kernel is one dimensional and so encodes one bit as either $(0,0,0)$ or $(1, 1, 1).$ If there is an error in the message $v\in F_2^3,$ perhaps a single bit was flipped, then we can hope to diagnose and recover the original message by computing $\partial v.$

So this is a way of viewing linear codes as "homologies". In this case we are using the homology of a circle. This would perhaps not be a very convincing application of homology theory except that it fits into a more general picture once we consider quantum codes. The prime example here is based on the homology of the torus, and is known as the "toric code". As with the repetition code above, we are working with a given cellular decomposition of the torus. This time we have a length two chain complex, still over the field $F_2$: $$V_2 \xrightarrow{\partial_2} V_1 \xrightarrow{\partial_1} V_0.$$ In this case the first homology group $ker(\partial_1)/im(\partial_2)$ is two dimensional and so encodes two quantum bits (qubits.)

Error processes act on $V_1$ (by flipping bits) and are diagnosed by $\partial_1.$ The error correction procedure is now a matter of finding the most likely homology class corresponding to the error. The topology of the situation is quite prominent: errors act locally on the torus cellulation, but can only effect the encoded information by percolating around the torus.

The dual chain complex, which we get by taking the transpose of the operators $\partial_2, \partial_1$ is also important, and a similar error correction procedure takes place on this dual complex.

Amazingly, various research groups around the world are presently engaged in actually building such things, as they hold great promise for robust storage of quantum information, which will be needed for any serious attempt at building a quantum computer.

(I should note that I have left out an explanation of how to go from the complex Hilbert spaces where qubits actually live to $F_2$ linear algebra. This is covered in the literature on quantum stabilizer codes.)

I am surprised no-one has mentioned Persistent Homology.

• en.wikipedia.org/wiki/… – Alexander Chervov Jun 20 '12 at 12:40
• See also José Figueroa-O'Farrill's comment on the accepted answer, in which he mentions a talk by Gunnar Carlsson about persistent homology. – J W Jun 20 '12 at 15:59

There are some applications of topology/cohomology to combinatorics and combinatoric geometry. One of the earliest examples is surely Lovasz's proof of a bound for the chromatic number of the Kneser graph; he uses the Borsuk-Ulam theorem, which is usually proved by homological methods. A modern exposition can be found here.

Another example is Tveberg's theorem with all its variants on the configuration of points in space (the best results can be found in a recent paper of Blagojevic, Matschke and Ziegler. There are many other results in convex geometry/polytope theory which use topological methods and, in particular, cohomology.

• While this isn't so clear from the title of the question, the first sentence in the text of the question mentions wanting things outside of pure mathematics. – tweetie-bird Oct 25 '12 at 14:27

An application of cohomology to provide a geometric/topological description charged particles in General Relativity can be found in

GRAVITATION: An Introduction to Current Research, ed. Louis Witten

and references therein.

Wheeler's geometrodynamics program contained a subprogram named "charge without charge", which aimed to express the electric charge in terms of geometric and/or topological properties. A wormhole allows the existence of an electromagnetic field without source, which looks like having sources - hence the name "charge without charge". The two ends of the wormholes behave as particles of opposite electric charge. And all this can be obtained as a solution to Einstein-Maxwell equations. Roots of the approach of Misner and Wheeler can be found in the paper of Einstein and Rosen, and a series of papers of G. Y. Rainich from 1924-1925.