Editing Homology (mathematics) (section)

== Homology of a topological space ==
Perhaps the most familiar usage of the term homology is for the ''homology of a topological space''. For sufficiently nice topological spaces and compatible choices of coefficient rings, any homology theory satisfying the [[Eilenberg–Steenrod axioms|Eilenberg-Steenrod axioms]] yields the same homology groups as the [[singular homology]] (see below) of that topological space, with the consequence that one often simply refers to the "homology" of that space, instead of specifying which homology theory was used to compute the homology groups in question.

For 1-dimensional topological spaces, probably the simplest homology theory to use is [[graph homology]], which could be regarded as a 1-dimensional special case of [[simplicial homology]], the latter of which involves a decomposition of the topological space into [[Simplex|simplices]]. (Simplices are a generalization of triangles to arbitrary dimension; for example, an edge in a graph is [[Homeomorphism|homeomorphic]] to a one-dimensional simplex, and a triangle-based pyramid is a 3-simplex.) Simplicial homology can in turn be generalized to [[singular homology]], which allows more general maps of simplices into the topological space. Replacing simplices with disks of various dimensions results in a related construction called [[cellular homology]].

There are also other ways of computing these homology groups, for example via [[Morse homology]], or by taking the output of the [[Universal coefficient theorem|Universal Coefficient Theorem]] when applied to a cohomology theory such as [[Čech cohomology]] or (in the case of real coefficients) [[De Rham cohomology]].

=== Inspirations for homology (informal discussion) ===
One of the ideas that led to the development of homology was the observation that certain low-dimensional shapes can be [[Topology|topologically]] distinguished by examining their "holes." For instance, a figure-eight shape has more holes than a circle <math>S^1</math>, and a 2-torus <math>T^2</math> (a 2-dimensional surface shaped like an inner tube) has different holes from a 2-sphere <math>S^2</math> (a 2-dimensional surface shaped like a basketball).

Studying topological features such as these led to the notion of the ''cycles'' that represent homology classes (the elements of homology groups). For example, the two [[Embedding|embedded]] circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus <math>T^2</math> and 2-sphere <math>S^2</math> represent 2-cycles. Cycles form a group under the operation of ''formal addition,'' which refers to adding cycles symbolically rather than combining them geometrically. Any formal sum of cycles is again called a cycle.

=== Cycles and boundaries (informal discussion) ===
Explicit constructions of homology groups are somewhat technical. As mentioned above, an explicit realization of the homology groups <math>H_n(X)</math> of a topological space <math>X</math> is defined in terms of the ''cycles'' and ''boundaries'' of a ''[[chain complex]]''  <math>  (C_\bullet, d_\bullet)</math> associated to <math>X</math>, where the type of chain complex depends on the choice of homology theory in use. These cycles and boundaries are elements of [[Abelian group|abelian groups]], and are defined in terms of the boundary homomorphisms <math>d_n: C_n \to C_{n-1}</math> of the chain complex, where each <math>C_n</math> is an abelian group, and the <math>d_n</math> are [[Group homomorphism|group homomorphisms]] that satisfy <math>d_{n-1} \circ d_n=0</math> for all <math>n</math>.

Since such constructions are somewhat technical, informal discussions of homology sometimes focus instead on topological notions that parallel some of the group-theoretic aspects of cycles and boundaries.

For example, in the context of [[Chain complex|chain complexes]], a '''boundary''' is any element of the [[Image (mathematics)|image]] <math>B_n := \mathrm{im}\, d_{n+1} :=\{d_{n+1}(c)\,|\; c\in C_{n+1}\}</math> of the boundary homomorphism <math>d_n: C_n \to C_{n-1}</math>, for some <math>n</math>. In topology, the boundary of a space is technically obtained by taking the space's [[Closure (topology)|closure]] minus its [[Interior (topology)|interior]], but it is also a notion familiar from examples, e.g., the boundary of the unit disk is the unit circle, or more topologically, the boundary of <math>D^2</math> is <math>S^1</math>.

Topologically, the boundary of the closed interval <math>[0,1]</math> is given by the disjoint union <math>\{0\} \, \amalg \, \{1\} </math>, and with respect to suitable orientation conventions, the oriented boundary of <math>[0,1]</math> is given by the union of a positively-oriented <math>\{1\} </math> with a negatively oriented <math>\{0\}. </math> The [[Simplicial homology|simplicial chain complex]] analog of this statement is that <math>d_1([0,1]) = \{1\} - \{0\} </math>. (Since <math>d_1 </math> is a homomorphism, this implies <math>d_1(k\cdot[0,1]) = k\cdot\{1\} - k\cdot\{0\} </math> for any integer <math>k </math>.)

In the context of chain complexes, a '''cycle''' is any element of the [[Kernel (category theory)|kernel]]<math>Z_n := \ker d_n :=\{c \in C_n \,|\; d_n(c) = 0\}</math>, for some <math>n</math>. In other words, <math>c \in C_n</math> is a cycle if and only if <math>d_n(c) = 0</math>. The closest topological analog of this idea would be a shape that has "no boundary," in the sense that its boundary is the empty set. For example, since <math>S^1, S^2  </math>, and <math>T^2 </math> have no boundary, one can associate cycles to each of these spaces. However, the chain complex notion of cycles (elements whose boundary is a "zero chain") is more general than the topological notion of a shape with no boundary.

It is this topological notion of no boundary that people generally have in mind when they claim that cycles can intuitively be thought of as detecting holes. The idea is that for no-boundary shapes like <math>S^1</math>, <math>S^2</math>,  and <math>T^2</math>, it is possible in each case to glue on a larger shape for which the original shape is the boundary. For instance, starting with a circle <math>S^1</math>, one could glue a 2-dimensional disk <math>D^2</math> to that <math>S^1</math> such that the <math>S^1</math> is the boundary of that <math>D^2</math>. Similarly, given a two-sphere <math>S^2</math>, one can glue a ball <math>B^3</math> to that <math>S^2</math> such that the <math>S^2</math> is the boundary of that <math>B^3</math>. This phenomenon is sometimes described as saying that <math>S^2</math> has a <math>B^3</math>-shaped "hole" or that it could be "filled in" with a <math>B^3</math>.

More generally, any shape with no boundary can be "filled in" with a [[Cone (topology)|cone]], since if a given space <math>Y</math> has no boundary, then the boundary of the cone on <math>Y
</math> is given by <math>Y</math>, and so if one "filled in" <math>Y</math> by gluing the cone on <math>Y</math> onto <math>Y</math>, then <math>Y</math> would be the boundary of that cone. (For example, a cone on <math>S^1</math> is [[Homeomorphism|homeomorphic]] to a disk <math>D^2</math> whose boundary is that <math>S^1</math>.) However, it is sometimes desirable to restrict to nicer spaces such as [[Manifold|manifolds]], and not every cone is homeomorphic to a manifold. [[Embedding|Embedded]] representatives of 1-cycles, 3-cycles, and oriented 2-cycles all admit manifold-shaped holes, but for example the real [[projective plane]] <math>\mathbb{RP}^2</math> and complex projective plane <math>\mathbb{CP}^2</math> have nontrivial [[cobordism]] classes and therefore cannot be "filled in" with manifolds.

On the other hand, the boundaries discussed in the homology of a topological space <math>X</math> are different from the boundaries of "filled in" holes, because the homology of a topological space <math>X</math> has to do with the original space <math>X</math>, and not with new shapes built from gluing extra pieces onto <math>X</math>. For example, any embedded circle <math>C</math> in <math>S^2</math> already bounds some embedded disk <math>D</math> in <math>S^2</math>, so such <math>C</math> gives rise to a boundary class in the homology of <math>S^2</math>. By contrast, no [[embedding]] of <math>S^1</math> into one of the 2 lobes of the figure-eight shape gives a boundary, despite the fact that it is possible to glue a disk onto a figure-eight lobe.

=== Homology groups ===
Given a sufficiently-nice topological space <math>X</math>, a choice of appropriate homology theory, and a chain complex <math>  (C_\bullet, d_\bullet)</math> associated to <math>X</math> that is compatible with that homology theory, the <math>n</math>th homology [[Group (mathematics)|group]] <math>H_n(X)</math> is then given by the [[quotient group]] <math>H_n(X)=Z_n/B_n</math> of <math>n</math>-cycles (<math>n</math>-dimensional cycles) modulo <math>n</math>-dimensional boundaries. In other words, the elements of <math>H_n(X)</math>, called ''homology classes'', are [[Equivalence class|equivalence classes]] whose representatives are <math>n</math>-cycles, and any two cycles are regarded as equal in <math>H_n(X)</math> if and only if they differ by the addition of a boundary. This also implies that the "zero" element of <math>H_n(X)</math> is given by the group of <math>n</math>-dimensional boundaries, which also includes formal sums of such boundaries.