Rp2 Homology
Nayme Cutforth
The similarity of a structure or function of parts of different origins based on their descent from a common evolutionary ancestor is homology. Analogy, by contrast, is a functional similarity of structure that is based on mere similarity of use. For example, the forelimbs of humans, bats, and deer are homologous; the form of construction and the number of bones in each are almost identical and represent adaptive modifications of the forelimb structure of their shared ancestor. The wings of birds and insects, on the other hand, are merely analogous; both are used for flight, but they do not share a common ancestral origin.
In mathematics, homology[a] is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.
The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold. Loosely speaking, a cycle is a closed submanifold, a boundary is a cycle which is also the boundary of a submanifold, and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries. A homology class is thus represented by a cycle which is not the boundary of any submanifold: the cycle represents a hole, namely a hypothetical manifold whose boundary would be that cycle, but which is "not there".
There are many different homology theories. A particular type of mathematical object, such as a topological space or a group, may have one or more associated homology theories. When the underlying object has a geometric interpretation as topological spaces do, the nth homology group represents behavior in dimension n. Most homology groups or modules may be formulated as derived functors on appropriate abelian categories, measuring the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a derived category.
Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic.[1] This was followed by Riemann's definition of genus and n-fold connectedness numerical invariants in 1857 and Betti's proof in 1871 of the independence of "homology numbers" from the choice of basis.[2]
On the ordinary sphere S 2 \displaystyle S^2 , the cycle b in the diagram can be shrunk to the pole, and even the equatorial great circle a can be shrunk in the same way. The Jordan curve theorem shows that any arbitrary cycle such as c can be similarly shrunk to a point. All cycles on the sphere can therefore be continuously transformed into each other and belong to the same homology class. They are said to be homologous to zero. Cutting a manifold along a cycle homologous to zero separates the manifold into two or more components. For example, cutting the sphere along a produces two hemispheres.
This is not generally true of cycles on other surfaces. The torus T 2 \displaystyle T^2 has cycles which cannot be continuously deformed into each other, for example in the diagram none of the cycles a, b or c can be deformed into one another. In particular, cycles a and b cannot be shrunk to a point whereas cycle c can, thus making it homologous to zero.
The edges of the square may then be glued back together in different ways. The square can be twisted to allow edges to meet in the opposite direction, as shown by the arrows in the diagram. The various ways of gluing the sides yield just four topologically distinct surfaces:
K 2 \displaystyle K^2 is the Klein bottle, which is a torus with a twist in it (In the square diagram, the twist can be seen as the reversal of the bottom arrow). It is a theorem that the re-glued surface must self-intersect (when immersed in Euclidean 3-space). Like the torus, cycles a and b cannot be shrunk while c can be. But unlike the torus, following b forwards right round and back reverses left and right, because b happens to cross over the twist given to one join. If an equidistant cut on one side of b is made, it returns on the other side and goes round the surface a second time before returning to its starting point, cutting out a twisted Mbius strip. Because local left and right can be arbitrarily re-oriented in this way, the surface as a whole is said to be non-orientable.
A square is a contractible topological space, which implies that it has trivial homology. Consequently, additional cuts disconnect it. The square is not the only shape in the plane that can be glued into a surface. Gluing opposite sides of an octagon, for example, produces a surface with two holes. In fact, all closed surfaces can be produced by gluing the sides of some polygon and all even-sided polygons (2n-gons) can be glued to make different manifolds. Conversely, a closed surface with n non-zero classes can be cut into a 2n-gon. Variations are also possible, for example a hexagon may also be glued to form a torus.[4]
A manifold with boundary or open manifold is topologically distinct from a closed manifold and can be created by making a cut in any suitable closed manifold. For example the disk or 2-ball B 2 \displaystyle B^2 is bounded by a circle S 1 \displaystyle S^1 . It may be created by cutting a trivial cycle in any 2-manifold and keeping the piece removed, by piercing the sphere and stretching the puncture wide, or by cutting the projective plane. It can also be seen as filling-in the circle in the plane.
When two cycles can be continuously deformed into each other, then cutting along one produces the same shape as cutting along the other, up to some bending and stretching. In this case the two cycles are said to be homologous or to lie in the same homology class. Additionally, if one cycle can be continuously deformed into a combination of other cycles, then cutting along the initial cycle is the same as cutting along the combination of other cycles. For example, cutting along a figure 8 is equivalent to cutting along its two lobes. In this case, the figure 8 is said to be homologous to the sum of its lobes.
This geometric analysis of manifolds is not rigorous. In a search for increased rigour, Poincar went on to develop the simplicial homology of a triangulated manifold and to create what is now called a chain complex.[6][7] These chain complexes (since greatly generalized) form the basis for most modern treatments of homology.
In such treatments a cycle need not be continuous: a 0-cycle is a set of points, and cutting along this cycle corresponds to puncturing the manifold. A 1-cycle corresponds to a set of closed loops (an image of the 1-manifold S 1 \displaystyle S^1 ). On a surface, cutting along a 1-cycle yields either disconnected pieces or a simpler shape. A 2-cycle corresponds to a collection of embedded surfaces such as a sphere or a torus, and so on.
The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".[11] Algebraic homology remains the primary method of classifying manifolds.[12]
A one-dimensional sphere S 1 \displaystyle S^1 is a circle. It has a single connected component and a one-dimensional-boundary hole, but no higher-dimensional holes. The corresponding homology groups are given as H k ( S 1 ) = { Z k = 0 , 1 0 otherwise \displaystyle H_k\left(S^1\right)=\begincases\mathbb Z &k=0,1\\\0\&\textotherwise\endcases where Z \displaystyle \mathbb Z is the group of integers and 0 \displaystyle \0\ is the trivial group. The group H 1 ( S 1 ) = Z \displaystyle H_1\left(S^1\right)=\mathbb Z represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.[14]
A two-dimensional sphere S 2 \displaystyle S^2 has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes. The corresponding homology groups are[14][15] H k ( S 2 ) = { Z k = 0 , 2 0 otherwise \displaystyle H_k\left(S^2\right)=\begincases\mathbb Z &k=0,2\\\0\&\textotherwise\endcases
A two-dimensional ball B 2 \displaystyle B^2 is a solid disc. It has a single path-connected component, but in contrast to the circle, has no higher-dimensional holes. The corresponding homology groups are all trivial except for H 0 ( B 2 ) = Z \displaystyle H_0\left(B^2\right)=\mathbb Z . In general, for an n-dimensional ball B n , \displaystyle B^n, [14]
The torus is defined as a product of two circles T 2 = S 1 S 1 \displaystyle T^2=S^1\times S^1 . The torus has a single path-connected component, two independent one-dimensional holes (indicated by circles in red and blue) and one two-dimensional hole as the interior of the torus. The corresponding homology groups are[16] H k ( T 2 ) = { Z k = 0 , 2 Z Z k = 1 0 otherwise \displaystyle H_k(T^2)=\begincases\mathbb Z &k=0,2\\\mathbb Z \times \mathbb Z &k=1\\\0\&\textotherwise\endcases
The different types of homology theory arise from functors mapping from various categories of mathematical objects to the category of chain complexes. In each case the composition of the functor from objects to chain complexes and the functor from chain complexes to homology groups defines the overall homology functor for the theory.[22]
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