Editing Homology (mathematics) (section)

=== 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.