What is a homology?

Comments (21)

1. Having just consulted my mathematical dictionary I fear that the answer to your request is simply, no! I hope I’m wrong and that one of your readers in capable of explaing it in comparatively non-mathematical terms but I doubt it.

   Come on folks prove me wrong!

2. I am not a mathematician, but this is my take on it:

   The basic idea of homology is to give a systematic way of classifying topological spaces. The simplest way of distinguishing spaces is by counting the number of holes – that’s the way that, as the lame joke has it, that a topologist can’t tell the difference between a coffee cup and a donut – they both have one hole. But that simple counting is not good enough to distinguish important differences between spaces – there can be different kinds of holes. So a better way of doing it is to assign a certain mathematical “group” to the space, and the way of assigning groups to spaces is “homology”.

3. The term is used in two completely different senses. In topology they use “homology groups”;
   these provide a coarse summary of information
   about the shape of an object.

   In geometry it refers to a symmetry of space
   (collineation) that fixes all lines on some point
   and all points on a line off the first point. (“Scaling everything up by a constant factor” would be an example.)

   1. That geometric definition actually makes sense, possibly because I can visualise it. Thanks.

4. It’s been a while, but I do like to say that I studied low-dimensional topology before I became a philosopher, so let’s see if I can help.

   Strictly speaking, a homology is just any sequence of groups connected by homomorphisms that satisfy some formal conditions. Topologists work with homologies and methods of constructing them that encode information about the structure of topological spaces.

   In the simplest construction, the different groups encode information about (very roughly!) the distinct ways of embedding spheres of different dimensions into the space. (A one-dimensional sphere is a circle; a two-dimensional sphere is a spherical surface as we normally think of it; a three-dimensional sphere is the surface of a four-dimensional ball; and so on.) For a given space, say the sequence of groups goes
   … -> G_3 -> G_2 -> G_1 -> G_0

   The group structure of G_0 encodes the number of distinct ways of embedding a 0-sphere into the space; it turns out that this is the same as the number of distinct connected parts of the space. G_1 tells you how many distinct ways you can embed a circle into the space. If the space is a 2-sphere (like the surface of the Earth), this is trivial, because all embedded circles can be collapsed down to a single point — imagine a rubber band around the equator, and gradually rolling it down to a single point at one of the poles. If the space is a torus (like the surface of a donut), there are two distinct ways: you can wrap a long circle all the way around the outside, or you can wrap a short circle through the donut hole. These two circles can’t be pushed or pulled in a smooth way to turn them into each other, or to get them to collapse down to a point.

   From G_2 on, things get more difficult to visualize. Which is, after all, why topologists invented homologies: the algebraic representation of the sequence of groups is much easier to work with than trying to imagine bending and stretching 5-spheres in 8-space or whatever. But the representation still works by representing distinct ways of embedding spheres into the space.

   That was probably still way too abstract, but I’ll check back in throughout the day and try to answer questions. 🙂

   1. I was wondering if you were going to drop in and take a crack at this.

   2. Dan, that is pitched about two levels of familiarity with mathematics higher than I current occupy. But thanks for the effort.

      1. You might try to find a topologist who can explain it in person, then. Sketches, clay, bits of string, and evocative gesturing are tools of serious research and teaching in my old mathematical home, but they don’t really have versions of those for blogs yet. Try the folks with the weird drawings and 17 different colors of chalk! Good luck!

         1. Thanks. I was hoping to avoid direct contact with topologists, having had some… interesting… encounters when I worked at Monash University, but I guess I have no alternative.

            And I thought philosophy was jargon-ridden…

5. Well, I think Mr. Hicks confuses homology and homotopy. They are related though.
   I don’t know any way to simply explain it. Furthermore there are different kinds of homology, were you thinking about something like singular cohomology? DeRham Cohomology?

6. There are several distinct meanings of homology:
   * A sequence of objects in a category. Usually, but not always, the category is taken to be that of groups. G_n->G_(n-1)->….->G_1->G_0. This sequence is a homology if the Image of one mapping is contained in the kernel of the other. In other words, if taking an object through two mappings arrives at identity.
   * Given a topological space, there are several ways to define a homology on it. The simplest is: Let G_n be the formal Z-based linear combinations of functions from an n-simplex (all points in n-space with co-ordinates which are positive and sum to no more than 1) into the space, where Z is the ring of whole numbers. Notice that the n-simplex contains n n-1 simplexes by setting the i-th co-ordinate to 0. Let f:G_n->G_n-1 be defined by f(function from simplex to space) = sum of -1**i * function from i-th n-1-sub-simplex. This defines a homology according to the previous definition. The structure of the Image of one function/Kernel of the previous are a classic toplogical invariant of the space, used to distinguish between different spaces. For example, this is an easy way to distinguish between the n-ball and the n-ball-without-0, proving that indeed they are different topological spaces.

   Regarding connection to homotopy theory: the homology Image/Kernel are related in structure to the homotopy groups (for example, the first non-zero one is the same in both, up to abelizations).

   Relation to Co-homology: It is possible to prove that if we are dealing with differential manifolds, requiring the functions to be differentiable rather than continuous does not change the homology structure. If we take the homology structure and tensor it with the real numbers we get the “non-twisted structure”, an important invariant of the homology structure. That structure is dual to the DeRham cohomology, in that n-forms define functionals on the n-simplex-functions, and the structures are indeed dual (which among other things, proves that if the homology is the same, the cohomology is the same). Note that because of the R-tensoring, the DeRham cohomology is unable to detect some structures that the homology detects (roughly, finite twists).

   This is all, of course, extremely sketchy and hand-wavy 🙂

   1. That is sketchy? Ohmigod, it’s worse than I thought…

7. David Williams at the NHM pointed out this passage for me:

   The corresponding parts in different animals being thus made namesakes, are called technically ‘homologues.’ The term is used by logicians as synonymous with ‘homonyms,’ and by geometricians as signifying ‘the sides of similar figures which are opposite to equal and corresponding angles,’ or to parts having the same proportions…

   from Richard Owens’ On the archetype and homologies of the vertebrate skeleton, 1848, page 5.

   1. HOMOLOGOUS, adj. homologous elements (such as terms, points, lines, angles). Elements that play similar roles in distinct figures or functions. The numerators or the denominators of two equal fractions are homologous terms. The vertices of a polygon and those of a projection of the polygon on a plane are homologous points and the sides and their projections are homologous lines. Syn. corresponding.

      James and James, Mathematical Dictionary, 4th Ed., Van Nostrand Reinhold Company, New York etc., 1976.

      1. Now that makes sense!

      2. You originally asked the wrong question 😉

      3. Topology didn’t exist when Owen’s wrote the above so for him a homolgy is just simply a one to one correspondence, which is all it really is when applied to topological sets but with a lot of complex conditions that define what exactly is meant by a one to one correspondence.

8. If you want to have a chat about it in person, I’m back in Brissie for a while as of tomorrow. I’m a total dumb amateur who knows only as much math as I need to make cool models of polyhedra out of paper, but Dan Hicks and moshez were helpful enough that I think I could get the idea across with chalk, coffee cups, donuts, and dramatic hand gestures.

   1. I do know a dorky joke about why mathematicians who study non-Abelian groups have to live on campus, which is a start. Maybe?

      1. Commuter jokes. Nice.

   2. Sounds great! I’ll supply the donuts, coffee and chalk, and you can supply the hands.

      I’ll be in at UQ tomorrow for the talk if you are going to be there.

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