Изворни код странице Homologija (matematika)

{{druge upotrebe-lat|Homologija (višeznačna odrednica)}}

U [[mathematics|matematici]], '''homologija'''<ref>in part from [[Greek language|Greek]] ὁμός ''homos'' "identical"</ref> je opšti način povezivanja niza algebarskih objekata kao što su [[Абелова група|abelove grupe]] ili [[module (mathematics)|moduli]] sa drugim matematičkim objektima kao što su [[topološki prostor]]i. Homološke grupe su prvobitno definisane u [[algebraic topology|algebarskoj topologiji]]. Slične konstrukcije su dostupne u širokom spektru drugih konteksta, kao što su [[apstraktna algebra]],<ref>{{Cite book |last1=Finston |first1=David R. |url=https://books.google.com/books?id=rLZjBAAAQBAJ |title=Abstract Algebra: Structure and Application |last2=Morandi |first2=Patrick J. |date=29 August 2014 |publisher=Springer |isbn=978-3-319-04498-9 |page=58 |language=en |quote=Much of our study of abstract algebra involves an analysis of structures and their operations}}</ref> [[Група (математика)|grupe]], [[Lijeva algebra]],<ref>{{cite book |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=Lie Groups and Lie Algebras: Chapters 1-3 |year=1989 |isbn=978-3-540-64242-8 |publisher=Springer |url=https://books.google.com/books?id=brSYF_rB2ZcC}}</ref><ref>{{Cite book |last1=Erdmann |first1=Karin |author-link=Karin Erdmann |last2=Wildon |first2=Mark |title=Introduction to Lie Algebras |publisher=Springer |year=2006 |isbn=1-84628-040-0}}</ref><ref>{{cite book |last=Hall |first=Brian C. |title=Lie groups, Lie algebras, and Representations: An Elementary Introduction |edition=2nd |series=Graduate Texts in Mathematics |volume=222 |publisher=Springer |year=2015 |isbn=978-3319134666 |doi=10.1007/978-3-319-13467-3 |issn=0072-5285}}</ref> [[Galois theory|teorija Galoa]] i [[algebarska geometrija]].<ref>{{citation|title=Invitation to Combinatorial Topology| first1=Maurice|last1=Fréchet|author-link1=Maurice René Fréchet| first2=Ky|last2=Fan|author-link2=Ky Fan|publisher=Courier Dover Publications|year=2012|isbn=9780486147888|page=101|url=https://books.google.com/books?id=dfLSzs0vHNQC&pg=PA101}}.</ref><ref>{{citation|title=A Combinatorial Introduction to Topology|first=Michael|last=Henle|publisher=Courier Dover Publications|year=1994|isbn=9780486679662|page=221|url=https://books.google.com/books?id=S0RJYkBxowEC&pg=PA221}}.</ref><ref>{{citation|title=Blowups, slicings and permutation groups in combinatorial topology|first=Jonathan|last=Spreer|publisher=Logos Verlag Berlin GmbH|year=2011|isbn=9783832529833|page=23|url=https://books.google.com/books?id=vdt5c07k0M4C&pg=PA23}}.</ref>

Prvobitna motivacija za definisanje grupa homologije bila je opservacija da se dva oblika mogu razlikovati putem ispitivanja njihovih otvora. Na primer, krug nije disk, jer krug ima otvor kroz njega dok je disk pun, a obična sfera nije krug, jer sfera okružuje dvodimenzionalni otvor, dok krug okružuje jednodimenzionalni otvor. Međutim, pošto je otvor „ne postoji”, nije odmah očigledno kako definisati otvor ili kako razlikovati različite vrste otvora. Homologija je izvorno bila rigorozna matematička metoda za definiranje i kategorizaciju otvora u [[mnogostrukost]]i. Slobodno govoreći, ciklus je zatvorena podmnogostrukost, ''granica'' je ciklus koji je takođe granica podmnogostrukosti, a ''klasa homologije'' (koja predstavlja otvor) je ekvivalentna klasi ciklusa po modularnim granicama. Klasa homologije je stoga predstavljena ciklusom koji nije granica bilo koje podmnogostrukosti: ciklus predstavlja otvor, odnosno hipotetičnu mnogostrukost čija bi granica bila taj ciklus, ali koji „nije tamo”.

Postoji mnogo različitih teorija homologije. Određeni tip matematičkog objekta, kao što je topološki prostor ili [[Група (математика)|grupa]], može imati jednu ili više povezanih teorija homologije. Kada osnovni objekat ima geometrijsku interpretaciju kao topološki prostori, -{''n''}--ta grupa homologije predstavlja ponašanje u dimenziji -{''n''}-. Većina grupa ili modula homologije moge se formulisati kao [[derived functor|izvedeni funktori]] na odgovarajućim [[abelian category|Abelovskim kategorijama]], merenjem neuspeha jednog funktora da bude [[exact functor|tačan]]. Iz ove apstraktne perspektive, grupe homologije se određuju objektima [[derived category|izvedene kategorije]].

== Pozadina ==

=== Poreklo ===

Smatra se da je teorija homologije nastala sa Ojlerovom formulom poliedra, ili [[Euler characteristic|Ojlerovom karakteristikom]].<ref>{{harvnb|Stillwell|1993|p=170}}</ref> Tomo je sledela [[Bernhard Riemann|Rimanova]] definicija numeričkih invarijanti [[Genus (mathematics)|rodova]] i n-tostruke povezanosti iy 1857. godine i [[Enrico Betti|Betijev]] dokaz nezavisnosti „homoloških brojeva” od izbora baze iz 1871. godine.<ref>{{harvnb|Weibel|1999|pp=2–3 (in PDF)}}</ref>

Sama homologija je razvijena kao način za analizu i klasifikaciju [[mnogostrukost]]i prema njihovim ''ciklusima'' - zatvorenim petljama (ili opštije podmnogostrukostima) koje se mogu nacrtati na datoj ''n''-dimenzionalnoj mnogostrukosti, ali ne i kontinuirano deformisanih jedne u druge.<ref name="Richeson254">{{harvnb|Richeson|2008|pp=254}}</ref> Ovi ciklusi se ponekad pominju i kao rezovi koji se mogu spojiti zajedno ili kao spojevi koji se mogu pričvrstiti i odvojiti. Ciklusi su klasifikovani po dimenzijama. Na primer, linija nacrtana na površini predstavlja 1-ciklus, zatvorenu petlju ili <math>S^1</math> (1-mnogostrukost), dok je površina prerezana kroz trodimenzionalnu mnogostrukost 2-ciklus.

=== Površine ===
[[Датотека:spherecycles1.svg|мини|лево|200px|Ciklusi na 2-sferi]]

Na običnoj [[Sfera|sferi]] <math>S^2</math>, ciklus ''b'' u dijagramu može se smanjiti do pola, a čak i ekvatorijalna [[velika kružnica]] ''a'' može se smanjiti na isti način. [[Jordan curve theorem|Teorema Žordanove krive]] pokazuje da se bilo koji proizvoljni ciklus, kao što je ''c'', može slično smanjiti do tačke. Svi ciklusi na sferi se stoga mogu kontinuirano transformisati jedan u drugi i pripadati istoj klasi homologije. Za njih se kaže da su homologni do nule. Presecanje mnogostrukosti duž ciklusa homolognog nuli razdvaja mnogostrukost na dve ili više komponenti. Na primer, sečenje sfere duž ''a'' proizvodi dve hemisfere.

[[Датотека:toruscycles1.svg|мини|256px|Ciklusi na torusu]]
Ovo se generalno ne odnosi na cikluse na drugim površinama. [[Torus]] <math>T^2</math> ima cikluse koji se ne mogu kontinuirano deformirati jedan u drugi, na primer u dijagramu ni jedan od ciklusa -{''a''}-, -{''b''}- ili -{''c''}- ne može biti deformisan jedan u drugi. Konkretno, ciklusi -{''a''}- i -{''b''}- se ne mogu smanjiti u tačku, dok ciklus -{''c''}- može, što ga čini homolognim na nulu.

Ako je površina torusa isečena duž oba ciklusa ''a'' i ''b'', ona se može otvoriti i spljoštiti u pravougaonik ili, još bolje, kvadrat. Jedan suprotan par strana predstavlja rez duž ''a'', a drugi suprotan par predstavlja rez duž ''b''.

Rubovi kvadrata mogu se zatim spojiti zajedno na različite načine. Kvadrat može biti zaokrenut da bi se ivice mogle susresti u suprotnom smeru, kao što je prikazano strelicama na dijagramu. U zavisnosti od simetrije, postoje četiri različita načina spajanja strana, od kojih svaka stvara različitu površinu:

[[Датотека:flatsurfaces.svg|мини|центар|460px|Četiri načina spajanja kvadrata da bi se napravila zatvorena površina: spojiti pojedinačne strelice i spojiti duple strelice.]]

== Vidi još ==
{{div col|colwidth=30em}}
* [[Betti number|Betijev broj]]
* [[Cycle space|Kružni prostor]]
* [[Eilenberg–Steenrod axioms|Ejlenberg-Stinrodovi aksiomi]]
* [[Cohomology|Kohomologija]]
* [[Homological algebra|Homološka algebra]]
* [[Homological conjectures in commutative algebra|Homološke pretpostavke u komutativnoj algebri]]
* [[Künneth theorem|Kinetova teorema]]
* [[List of cohomology theories|Spisak kohomoloških teorija]]
* [[De Rham cohomology|De Ramova kohomologija]]
{{div col end}}

== Reference ==
{{reflist}}

== Literatura ==
{{Литература|30em}}
* [[Henri Cartan|Cartan, Henri Paul]] and [[Samuel Eilenberg|Eilenberg, Samuel]] (1956) ''Homological Algebra'' Princeton University Press, Princeton, NJ, {{OCLC|529171}}
* Eilenberg, Samuel and Moore, J. C. (1965) ''Foundations of relative homological algebra'' (Memoirs of the American Mathematical Society number 55) American Mathematical Society, Providence, R.I., {{OCLC|1361982}}
* [http://www.encyclopediaofmath.org/index.php/Homology_group ''Homology group'' at Encyclopaedia of Mathematics]
* {{citation|last=Hilton|first=Peter|year=1988|title=A Brief, Subjective History of Homology and Homotopy Theory in This Century |journal = Mathematics Magazine|volume=60|issue=5|pages=282–291| jstor = 2689545|publisher=Mathematical Association of America|doi=10.1080/0025570X.1988.11977391 }}
* {{citation|title=The Heritage of Emmy Noether|editor-first= M.|editor-last =Teicher|editor-link= Mina Teicher |series=Israel Mathematical Conference Proceedings|publisher= [[Bar-Ilan University]]/[[American Mathematical Society]]/[[Oxford University Press]]|year= 1999|oclc= 223099225 |isbn= 978-0-19-851045-1}}
* Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topology'', Princeton University (2008)
* Spanier, Edwin H. (1966). ''Algebraic Topology'', Springer, p.&nbsp;155,. {{isbn|0-387-90646-0}}.
* Timothy Gowers, June Barrow-Green, Imre Leader (2010), ''[[The Princeton Companion to Mathematics]]'', Princeton University Press, {{isbn|9781400830398}}.
* {{cite book|doi=10.1007/978-1-4612-4372-4_6 |chapter=Homology Theory and Abelianization |title=Classical Topology and Combinatorial Group Theory |series=Graduate Texts in Mathematics |year=1993 |last1=Stillwell |first1=John |volume=72 |pages=169–184 |isbn=978-0-387-97970-0 |publisher=Springer}}
* Charles A. Weibel (1999), ''[http://www.math.uiuc.edu/K-theory/0245/survey.pdf History of Homological Algebra]'', chapter 28 in the book ''History of Topology'' by I.M. James, Elsevier, {{isbn|9780080534077}}.
* {{citation |first=Dylan G. L. |last=Allegretti |url=http://www.math.uchicago.edu/~may/VIGRE/VIGREREU2008.html |title=Simplicial Sets and van Kampen's Theorem |year=2008}} ''(Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets).''
* {{citation |last=Bredon |first=Glen E. |author-link=Glen Bredon|title=Topology and Geometry |year=1993 |publisher=Springer |series=Graduate Texts in Mathematics |volume=139 |url=https://books.google.com/books?id=G74V6UzL_PUC |isbn=0-387-97926-3}}.
* {{citation |first=R. |last=Brown |author-link=Ronald Brown (mathematician) |url=http://www.bangor.ac.uk/r.brown/hdaweb2.html |title=Higher dimensional group theory |year=2007 |accessdate=26. 06. 2023 |archive-date=14. 05. 2016 |archive-url=http://arquivo.pt/wayback/20160514115207/http://www.bangor.ac.uk/r.brown/hdaweb2.html |url-status= }} ''(Gives a broad view of higher-dimensional van Kampen theorems involving multiple groupoids)''.
* {{citation |first1=R. |last1=Brown |first2=A. |last2=Razak |title=A van Kampen theorem for unions of non-connected spaces |journal=Arch. Math. |volume=42 |pages=85–88 |year=1984 |doi=10.1007/BF01198133 |s2cid=122228464 }}. "Gives a general theorem on the [[fundamental groupoid]] with a set of base points of a space which is the union of open sets."
* {{citation |first1=R. |last1=Brown |first2=K. |last2=Hardie |first3=H. |last3=Kamps |first4=T. |last4=Porter |title=The homotopy double groupoid of a Hausdorff space |journal=Theory Appl. Categories |volume=10 |issue=2 |pages=71–93 |year=2002 |url=http://www.emis.de/journals/TAC/volumes/10/2/10-02abs.html}}.
* {{citation |first1=R. |last1=Brown |first2=P.J. |last2=Higgins |title=On the connection between the second relative homotopy groups of some related spaces |journal=Proc. London Math. Soc. |volume=S3-36 |issue= 2|pages=193–212 |year=1978 |doi=10.1112/plms/s3-36.2.193 }}. "The first 2-dimensional version of van Kampen's theorem." 
* {{citation |first1=Ronald |last1=Brown |first2=Philip J. |last2=Higgins |first3=Rafael |last3=Sivera |title=Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids |url=http://www.bangor.ac.uk/~mas010/nonab-a-t.html |year=2011 |publisher=European Mathematical Society |isbn=978-3-03719-083-8 |series=European Mathematical Society Tracts in Mathematics |volume=15|archive-url=https://web.archive.org/web/20090604050453/http://www.bangor.ac.uk/~mas010/nonab-a-t.html |archive-date=2009-06-04 |arxiv=math/0407275 }} This provides a homotopy theoretic approach to basic algebraic topology, without needing a basis in [[singular homology]], or the method of simplicial approximation. It contains a lot of material on [[crossed module]]s.
* {{ citation | first1 = John B. | last1 = Fraleigh | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
* {{citation |author-link1=Marvin Greenberg |last1=Greenberg |first1=Marvin J. |first2=John R. |last2=Harper |title=Algebraic Topology: A First Course, Revised edition |year=1981 |publisher=Westview/Perseus |series=Mathematics Lecture Note Series |isbn=9780805335576 |url-access=registration |url=https://archive.org/details/algebraictopolog00gree_0 }}. A functorial, algebraic approach originally by Greenberg with geometric flavoring added by Harper. 
* {{citation| last=Hatcher |first= Allen |author-link=Allen Hatcher| title=Algebraic Topology |url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html |year= 2002 |publisher=Cambridge University Press |place=Cambridge |isbn=0-521-79540-0}}. A modern, geometrically flavoured introduction to algebraic topology.
* {{citation |first=Philip J. |last=Higgins |title=Notes on categories and groupoids |url=https://books.google.com/books?id=IqdIAAAAMAAJ |year=1971 |publisher=Van Nostrand Reinhold |isbn=9780442034061}}
* {{citation| last=Maunder |first=C. R. F. |title=Algebraic Topology |year=1970 |publisher= Van Nostrand Reinhold |place=London |isbn=0-486-69131-4}}.
* {{citation |author-link=Tammo tom Dieck |first=Tammo |last=tom Dieck |title=Algebraic Topology |url=https://books.google.com/books?id=ruSqmB7LWOcC |year=2008 |publisher=European Mathematical Society |isbn=978-3-03719-048-7 |series=EMS Textbooks in Mathematics}}
* {{citation|first=Egbert|last= van Kampen|author-link=Egbert van Kampen| title=On the connection between the fundamental groups of some related spaces| journal=[[American Journal of Mathematics]]|volume= 55 |year=1933 |issue= 1|pages=261–7 |jstor=51000091}}
* {{cite book|first=Allen|last= Hatcher|author-link=Allen Hatcher| url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html |title=Algebraic topology|year=2002|publisher= [[Cambridge University Press]]|isbn=0-521-79160-X}} and {{ISBN|0-521-79540-0}}.
* {{Springer |title=Algebraic topology |id=p/a011700}}
* {{cite book |last=May |first=J. Peter |author-link=J. Peter May|title=A Concise Course in Algebraic Topology |year=1999 |publisher=[[University of Chicago Press]] |url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |archive-date=2022-10-09 |url-status=live |access-date=2008-09-27 }}
* {{Cite journal |last=Cayley |first=A. |date=1854 |title=On the theory of groups, as depending on the symbolic equation θ<sup>n</sup> = 1 |url=https://babel.hathitrust.org/cgi/pt?id=pst.000068485757;view=1up;seq=54 |journal=Philosophical Magazine |series=4th series |volume=7 |issue=42 |pages=40–47 |doi=10.1080/14786445408647421}}
* {{Cite book |last=Kronecker |first=Leopold |url=https://archive.org/details/leopoldkronecke00berlgoog/page/274/mode/2up?view=theater |title=Leopold Kronecker's werke : Herausgegeben auf veranlassung der Königlich preussischen akademie der wissenschaften |date=1895 |publisher=Leipzig ; Berlin : B.G. Teubner |editor-last=Hensel |editor-first=Kurt |page=275 |chapter=Auseinandeesetzung einiger eigenschaften der klassenanzahl idealer complexer zahlen |trans-chapter=An exposition of some properties of the class number of ideal complex numbers}}
* {{Cite journal |last=Frobenius |first=G. |date=April 2008 |title=Neuer Beweis des Sylowschen Satzes |trans-title=New Proof of Sylow's Theorem |url=http://mmsengineering.ca/sasha-web/resources/frobenius/gutfraind_Frobenius1887.pdf |journal=Journal für die reine und angewandte Mathematik |volume=1887 |issue=100 |pages=179–181 |doi=10.1515/crll.1887.100.179 |s2cid=117970003 |language=de |translator-first=Sasha |translator-last=Gutfraind}}
* {{cite book |last=Beltiţă |first= Daniel |title=Smooth Homogeneous Structures in Operator Theory |series=CRC Monographs and Surveys in Pure and Applied Mathematics |volume=137 |publisher=CRC Press |year= 2006 |isbn=978-1-4200-3480-6 |mr=2188389 |url=https://books.google.com/books?id=x8NQc-pWbLQC&q=%22Lie+algebra%22}}
* {{Cite journal |last1=Boza |first1=Luis |last2=Fedriani |first2=Eugenio M. |last3=Núñez |first3=Juan |date=2001-06-01 |title=A new method for classifying complex filiform Lie algebras |journal=Applied Mathematics and Computation |volume=121 |issue=2–3 |pages=169–175 |doi=10.1016/s0096-3003(99)00270-2 |issn=0096-3003}}
* {{cite book |last1=Hofmann |first1=Karl H. |last2=Morris |first2=Sidney A |title=The Lie Theory of Connected Pro-Lie Groups |publisher=European Mathematical Society |year=2007 |isbn=978-3-03719-032-6}}
* {{cite book |last=Humphreys |first=James E. |author-link=James E. Humphreys |title=Introduction to Lie Algebras and Representation Theory |edition=2nd |series=Graduate Texts in Mathematics |volume=9 |publisher=Springer-Verlag |year=1978 |isbn=978-0-387-90053-7 |url-access=registration |url=https://archive.org/details/introductiontoli00jame}}
* {{cite book |last=Jacobson |first=Nathan |author-link=Nathan Jacobson |title=Lie algebras |orig-year=1962 |publisher=Dover |year=1979 |isbn=978-0-486-63832-4 |ref={{harvid |Jacobson |1962}}}}
* {{cite book |last1=Kac |first1=Victor G. |author-link1=Victor Kac |display-authors=etal |title=Course notes for MIT 18.745: Introduction to Lie Algebras |url=http://math.mit.edu/~lesha/745lec/ |url-status=bot: unknown |archive-url=https://web.archive.org/web/20100420004313/http://math.mit.edu/~lesha/745lec/ |archive-date=2010-04-20}}
* {{cite journal |last1=Mubarakzyanov |first1=G.M. |title=On solvable Lie algebras |journal=Izv. Vys. Ucheb. Zaved. Matematika |volume=1 |issue=32 |year=1963 |pages=114–123 |mr=153714 |zbl=0166.04104 |url=http://mi.mathnet.ru/eng/ivm2141 |language=ru}}
* {{cite web |last1=O'Connor |first1=J.J |author-link1=John J. O'Connor (mathematician) |last2=Robertson |first2=E.F. |author-link2=Edmund F. Robertson |title=Biography of Sophus Lie |year=2000 |publisher=MacTutor History of Mathematics Archive |url=http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Lie.html}}
* {{cite web |last1=O'Connor |first1=J.J |last2=Robertson |first2=E.F. |title=Biography of Wilhelm Killing |year=2005 |publisher=MacTutor History of Mathematics Archive |url=http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Killing.html}}
* {{cite journal |last1=Popovych |first1=R.O. |last2=Boyko |first2=V.M. |last3=Nesterenko |first3=M.O. |last4=Lutfullin |first4=M.W. |display-authors=etal |title=Realizations of real low-dimensional Lie algebras |journal=J. Phys. A: Math. Gen. |volume=36 |issue=26 |year=2003 |pages=7337–60 |doi=10.1088/0305-4470/36/26/309 |arxiv=math-ph/0301029 |bibcode=2003JPhA...36.7337P |s2cid=9800361}}
* {{cite book |last=Serre |first= Jean-Pierre |author-link=Jean-Pierre Serre |title=Lie Algebras and Lie Groups |edition=2nd |publisher=Springer |year=2006 |isbn=978-3-540-55008-2}}
* {{cite book |last=Steeb |first= Willi-Hans |title=Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra |edition=2nd |publisher= World Scientific |year= 2007 |isbn=978-981-270-809-0 |mr=2382250 |doi=10.1142/6515}}
* {{cite book |last=Varadarajan |first=Veeravalli S. |author-link=Veeravalli S. Varadarajan |title=Lie Groups, Lie Algebras, and Their Representations |edition=1st |publisher=Springer |year=2004 |isbn=978-0-387-90969-1}}

{{Литература крај}}

== Spoljašnje veze ==
{{Commons category-lat|}}

{{L|Топологија}}
{{нормативна контрола-лат}}
{{Portal bar-lat|Matematika}}

[[Категорија:Теорија хомологије]]