Editing Homology (mathematics) (section)

=== Surfaces ===
{{multiple image
| align             = centre
| total_width       = 750
| image1            = spherecycles1.svg
| caption1          = Cycles on a 2-sphere <math>S^2</math>
| image2            = toruscycles1.svg
| caption2          = Cycles on a torus <math>T^2</math>
| image3            = Kleincycles1.svg
| caption3          = Cycles on a Klein{{br}}bottle <math>K^2</math>
| image4            = projectivecycles2.svg
| caption4          = Cycles on a hemispherical projective plane <math>P^2</math>
}}

On the ordinary [[sphere]] <math>S^2</math>, the curve ''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 closed curve such as ''c'' can be similarly shrunk to a point. This implies that <math>S^2</math> has trivial [[fundamental group]], so as a consequence, it also has trivial first homology group.

The [[torus]] <math>T^2</math> has closed curves 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.

If the torus surface is cut along both ''a'' and ''b'', it can be opened out and flattened into a rectangle or, more conveniently, a square. One opposite pair of sides represents the cut along ''a'', and the other opposite pair represents the cut along ''b''.

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:
{{Clear}}

[[File:flatsurfaces.svg|thumb|centre|460px|The four ways of gluing a square to make a closed surface: glue single arrows together and glue double arrows together so that the arrowheads point in the same direction.]]
<math>K^2</math> 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 [[Möbius 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.

The [[projective plane]] <math>P^2</math> has both joins twisted. The uncut form, generally represented as the [[Boy surface]], is visually complex, so a hemispherical embedding is shown in the diagram, in which antipodal points around the rim such as ''A'' and ''A&prime;'' are identified as the same point. Again, ''a'' is non-shrinkable while ''c'' is. If ''b'' were only wound once, it would also be non-shrinkable and reverse left and right. However it is wound a second time, which swaps right and left back again; it can be shrunk to a point and is homologous to ''c''.

Cycles can be joined or added together, as ''a'' and ''b'' on the torus were when it was cut open and flattened down. In the [[Klein bottle]] diagram, ''a'' goes round one way and &minus;''a'' goes round the opposite way. If ''a'' is thought of as a cut, then  −''a'' can be thought of as a gluing operation. Making a cut and then re-gluing it does not change the surface, so ''a'' + (−''a'') = 0.

But now consider two ''a''-cycles. Since the Klein bottle is nonorientable, you can transport one of them all the way round the bottle (along the ''b''-cycle), and it will come back as −''a''. This is because the Klein bottle is made from a cylinder, whose ''a''-cycle ends are glued together with opposite orientations. Hence 2''a'' = ''a'' + ''a'' = ''a'' + (−''a'') = 0. This phenomenon is called [[torsion (algebra)|torsion]]. Similarly, in the projective plane, following the unshrinkable cycle ''b'' round twice remarkably creates a trivial cycle which ''can'' be shrunk to a point; that is, ''b'' + ''b'' = 0. Because ''b'' must be followed around twice to achieve a zero cycle, the surface is said to have a torsion coefficient of 2. However, following a ''b''-cycle around twice in the Klein bottle gives simply ''b'' + ''b'' = 2''b'', since this cycle lives in a torsion-free homology class. This corresponds to the fact that in the fundamental polygon of the Klein bottle, only one pair of sides is glued with a twist, whereas in the projective plane both sides are twisted.

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 (2''n''-gons) can be glued to make different manifolds. Conversely, a closed surface with ''n'' non-zero classes can be cut into a 2''n''-gon. Variations are also possible, for example a hexagon may also be glued to form a torus.<ref name="weeks">{{cite book |last=Weeks |first=Jeffrey R. |url=https://books.google.com/books?id=ZlVwDwAAQBAJ |title=The Shape of Space |date=2001 |publisher=CRC Press |isbn=978-0-203-91266-9 }}</ref>

The first recognisable theory of homology was published by [[Henri Poincaré]] in his seminal paper "[[Analysis Situs (paper)|Analysis situs]]", ''J. Ecole polytech.'' (2) '''1'''. 1–121 (1895). The paper introduced homology classes and relations. The possible configurations of orientable cycles are classified by the [[Betti number]]s of the manifold (Betti numbers are a refinement of the Euler characteristic).  Classifying the non-orientable cycles requires additional information about torsion coefficients.<ref name="Richeson254">{{harvnb|Richeson|2008|page=254}}</ref>

The complete classification of 1- and 2-manifolds is given in the table.

{| class="wikitable sortable"
|+ Topological characteristics of closed 1- and 2-manifolds<ref name="Richeson">{{harvnb|Richeson|2008}}</ref>
|-
! colspan="2" | Manifold
! rowspan="2" | [[Euler characteristic|Euler no.]], {{br}}χ
! rowspan="2" | Orientability
! colspan="3" | [[Betti number]]s
! rowspan="2" | Torsion coefficient {{br}}(1-dimensional) 
|-
! Symbol<ref name="weeks" />
! Name
! style="width:3em;" | ''b''<sub>0</sub>
! ''b''<sub>1</sub>
! style="width:3em;" | ''b''<sub>2</sub>
|-
| <math>S^1</math> || [[Circle]] (1-manifold) || {{fsp}}0 || Orientable || 1 || 1 || {{N/A}} || {{N/A}}
|-
| <math>S^2</math> || [[Sphere]] || {{fsp}}2 || Orientable || 1 || 0 || 1 || None
|-
| <math>T^2</math> || [[Torus]] || {{fsp}}0 || Orientable || 1 || 2 || 1 || None
|-
| <math>P^2</math> || [[Projective plane]] || {{fsp}}1 || Non-orientable || 1 || 0 || 0 || 2
|-
| <math>K^2</math> || [[Klein bottle]] || {{fsp}}0 || Non-orientable || 1 || 1 || 0 || 2
|-
| || 2-holed torus || −2 || Orientable || 1 || 4 || 1 || None
|-
| || [[Genus g surface|''g''-holed torus]] (''g'' is the [[genus (topology)|genus]]) || {{fsp}}2 − 2''g'' || Orientable || 1 || 2''g'' || 1 || None
|-
| || Sphere with ''c'' [[cross-cap]]s || {{fsp}}2 − ''c'' || Non-orientable || 1 || ''c'' − 1 || 0 || 2
|-
| || 2-Manifold with ''g''{{nbsp}}holes and ''c''{{nbsp}}cross-caps (''c''{{nbsp}}>{{nbsp}}0)|| {{fsp}}2{{nbsp}}−{{nbsp}}(2''g''{{nbsp}}+{{nbsp}}''c'') || Non-orientable || 1 || (2''g''{{nbsp}}+{{nbsp}}''c''){{nbsp}}−{{nbsp}}1 || 0 || 2
|}
: Notes
:# For a non-orientable surface, a hole is equivalent to two cross-caps.
:# Any closed 2-manifold can be realised as the [[connected sum]] of ''g'' tori and ''c'' projective planes, where the 2-sphere <math>S^2</math> is regarded as the empty connected sum. Homology is preserved by the operation of connected sum.

In a search for increased rigour, Poincaré went on to develop the simplicial homology of a triangulated manifold and to create what is now called a simplicial [[chain complex]].<ref name="Richeson258">{{harvnb|Richeson|2008|page=258}}</ref><ref>{{Harvnb|Weibel|1999|p=4}}</ref>  Chain complexes (since greatly generalized) form the basis for most modern treatments of homology.

[[Emmy Noether]] and, independently, [[Leopold Vietoris]] and [[Walther Mayer]] further developed the theory of algebraic homology groups in the period 1925–28.<ref>{{Harvnb|Hilton|1988|p=284}}</ref><ref>For example [http://smf4.emath.fr/Publications/Gazette/2011/127/smf_gazette_127_15-44.pdf ''L'émergence de la notion de groupe d'homologie'', Nicolas Basbois (PDF)], in French, note 41, explicitly names Noether as inventing the homology group.</ref><ref>Hirzebruch, Friedrich, [http://www.mathe2.uni-bayreuth.de/axel/papers/hierzebruch:emmy_noether_and_topology.ps.gz Emmy Noether and Topology] in {{Harvnb|Teicher|1999|pp=61–63}}.</ref> The new [[combinatorial topology]] formally treated topological classes as [[abelian group]]s.  Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and in the special case of surfaces, the torsion part of the homology group only occurs for non-orientable cycles.

The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "[[algebraic topology]]".<ref>[http://math.vassar.edu/faculty/McCleary/BourbakiAlgTop.pdf ''Bourbaki and Algebraic Topology'' by John McCleary (PDF)] {{Webarchive|url=https://web.archive.org/web/20080723154154/http://math.vassar.edu/faculty/McCleary/BourbakiAlgTop.pdf|date=2008-07-23}} gives documentation (translated into English from French originals).</ref>  Algebraic homology remains the primary method of classifying manifolds.<ref>{{harvnb|Richeson|2008|page=264}}</ref>