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403:
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38:
601:
defined on the manifold in question. De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology. This was extended in the 1950s,
503:) of spaces. This allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure, often making these statements easier to prove. Two major ways in which this can be done are through
86:
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any
310:. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an
622:
of spaces passes to an isomorphism of homology groups), verified that all existing (co)homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.
285:
which cannot be embedded in three dimensions, but can be embedded in four dimensions. Typically, results in algebraic topology focus on global, non-differentiable aspects of manifolds; for example
488:, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces.
233:. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign "quantities" to the
378:
349:
2209:
125:, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
751:
is free. This result is quite interesting, because the statement is purely algebraic yet the simplest known proof is topological. Namely, any free group
384:); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
1690:
1542:
670:
484:). In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic
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on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.
535:
480:, implying an emphasis on how a space X was constructed from simpler ones (the modern standard tool for such construction is the
2187:
1704:
850:
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1213:
1149:
1129:
1010:
1522:
2089:
1778:
1184:
1164:
1504:
Brown, R.; Higgins, P.J. (1978), "On the connection between the second relative homotopy groups of some related spaces",
1144:
570:
have the same associated groups, but their associated morphisms also correspondâa continuous mapping of spaces induces a
1429:
1425:
855:
845:
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519:
groups. The fundamental groups give us basic information about the structure of a topological space, but they are often
1139:
2230:
1799:
1773:
1124:
636:
442:
31:
1114:
790:
is free. On the other hand, this type of application is also handled more simply by the use of covering morphisms of
2097:
1741:
1296:
1228:
441:
appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an
235:
801:
1818:
Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids.
1169:
2168:
1896:
714:
534:
Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated.
1119:
733:
2182:
1253:
677:
582:
528:
468:, but still retains a combinatorial nature that allows for computation (often with a much smaller complex).
65:
1396:(Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets).
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2112:
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905:
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1959:
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794:, and that technique has yielded subgroup theorems not yet proved by methods of algebraic topology; see
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282:
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354:
351:. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of
325:
2107:
2059:
2033:
1881:
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703:
when the top-dimensional integral homology group is the integers, and is non-orientable when it is 0.
164:
1609:. A functorial, algebraic approach originally by Greenberg with geometric flavoring added by Harper.
1553:
This provides a homotopy theoretic approach to basic algebraic topology, without needing a basis in
1527:, European Mathematical Society Tracts in Mathematics, vol. 15, European Mathematical Society,
2158:
1954:
1258:
1233:
1174:
1000:
640:
492:
485:
437:(see illustration). Simplicial complexes should not be confused with the more abstract notion of a
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2013:
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2006:
1964:
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61:
1524:
Nonabelian
Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids
1487:
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1920:
1866:
1587:
1509:
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1154:
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57:
1447:(Gives a broad view of higher-dimensional van Kampen theorems involving multiple groupoids)
1979:
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1453:
Brown, R.; Razak, A. (1984), "A van Kampen theorem for unions of non-connected spaces",
566:
of the underlying topological space, in the sense that two topological spaces which are
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2001:
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originated here. Fundamental groups and homology and cohomology groups are not only
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1984:
1930:
1786:
1702:(1933), "On the connection between the fundamental groups of some related spaces",
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1005:
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One of the first mathematicians to work with different types of cohomology was
2018:
1949:
1908:
1557:, or the method of simplicial approximation. It contains a lot of material on
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In mathematics, homotopy groups are used in algebraic topology to classify
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1996:
1945:
1852:
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707:
500:
491:
In the algebraic approach, one finds a correspondence between spaces and
427:
248:
198:
156:
88:
80:
1717:
1633:. A modern, geometrically flavoured introduction to algebraic topology.
1466:
611:
555:
523:
and can be difficult to work with. The fundamental group of a (finite)
431:
1482:
with a set of base points of a space which is the union of open sets."
1533:
1300:
315:
270:
610:
generalized this approach. They defined homology and cohomology as
1360:
Blowups, slicings and permutation groups in combinatorial topology
538:
are completely classified and are particularly easy to work with.
274:
73:
45:, one of the most frequently studied objects in algebraic topology
42:
36:
1263:
103:
Below are some of the main areas studied in algebraic topology:
37:
1825:
1685:, EMS Textbooks in Mathematics, European Mathematical Society,
277:, which can all be realized in three dimensions, but also the
1517:. "The first 2-dimensional version of van Kampen's theorem."
1821:
155:"identical") is a certain general procedure to associate a
1521:
Brown, Ronald; Higgins, Philip J.; Sivera, Rafael (2011),
1410:, Graduate Texts in Mathematics, vol. 139, Springer,
209:. That is, cohomology is defined as the abstract study of
744:-space identifies at least one pair of antipodal points.
546:
In general, all constructions of algebraic topology are
460:. This class of spaces is broader and has some better
1598:, Mathematics Lecture Note Series, Westview/Perseus,
1486:
Brown, R.; Hardie, K.; Kamps, H.; Porter, T. (2002),
357:
328:
631:
Classic applications of algebraic topology include:
418:
of a certain kind, constructed by "gluing together"
221:. Cohomology can be viewed as a method of assigning
2088:
2052:
1938:
1859:
1594:
Algebraic
Topology: A First Course, Revised edition
1488:"The homotopy double groupoid of a Hausdorff space"
786:is again a graph. Therefore, its fundamental group
1591:
372:
343:
774:is the fundamental group of some covering space
225:to a topological space that has a more refined
121:. The first and simplest homotopy group is the
30:For the topology of pointwise convergence, see
755:may be realized as the fundamental group of a
725: = 2, this is sometimes called the "
1837:
452:is a type of topological space introduced by
8:
581:. One can use the differential structure of
1336:, Courier Dover Publications, p. 221,
1309:, Courier Dover Publications, p. 101,
713:admits a nowhere-vanishing continuous unit
167:with a given mathematical object such as a
2205:
2178:
1844:
1830:
1822:
676:One can use the differential structure of
1622:, Cambridge: Cambridge University Press,
1532:
364:
360:
359:
356:
335:
331:
330:
327:
1391:Simplicial Sets and van Kampen's Theorem
1363:, Logos Verlag Berlin GmbH, p. 23,
1333:A Combinatorial Introduction to Topology
1283:
401:
1276:
795:
1792:A Concise Course in Algebraic Topology
64:. The basic goal is to find algebraic
79:, though usually most classify up to
7:
1306:Invitation to Combinatorial Topology
696:defined on the manifold in question.
669:, which allows one to calculate the
618:subject to certain axioms (e.g., a
1567:A First Course In Abstract Algebra
1478:. "Gives a general theorem on the
692:to investigate the solvability of
597:to investigate the solvability of
476:An older name for the subject was
25:
1661:, London: Van Nostrand Reinhold,
1640:Notes on categories and groupoids
536:Finitely generated abelian groups
2204:
2177:
2167:
2157:
2146:
2136:
2135:
1929:
1388:Allegretti, Dylan G. L. (2008),
1264:Topological quantum field theory
373:{\displaystyle \mathbb {R} ^{3}}
344:{\displaystyle \mathbb {R} ^{3}}
1809:from the original on 2022-10-09
1705:American Journal of Mathematics
1431:Higher dimensional group theory
261:that near each point resembles
1214:Glossary of algebraic topology
1150:Freudenthal suspension theorem
1130:Cellular approximation theorem
736:: any continuous map from the
495:that respects the relation of
472:Method of algebraic invariants
1:
1185:Universal coefficient theorem
1165:Lefschetz fixed-point theorem
766:tells us that every subgroup
671:EulerâPoincarĂ© characteristic
846:Luitzen Egbertus Jan Brouwer
650:to itself has a fixed point.
1800:University of Chicago Press
1774:Encyclopedia of Mathematics
1637:Higgins, Philip J. (1971),
1125:Brouwer fixed point theorem
637:Brouwer fixed point theorem
443:abstract simplicial complex
32:Algebraic topology (object)
2247:
2098:Banach fixed-point theorem
1742:Cambridge University Press
1657:Maunder, C. R. F. (1970),
1590:; Harper, John R. (1981),
1565:Fraleigh, John B. (1976),
1229:Higher-dimensional algebra
1180:Seifertâvan Kampen theorem
542:Setting in category theory
391:
296:
246:
182:
139:In algebraic topology and
132:
110:
29:
2131:
1927:
1643:, Van Nostrand Reinhold,
1569:(2nd ed.), Reading:
1357:Spreer, Jonathan (2011),
802:Topological combinatorics
435:-dimensional counterparts
380:upon itself (known as an
1514:10.1112/plms/s3-36.2.193
1175:Poincaré duality theorem
1145:EilenbergâZilber theorem
197:is a general term for a
193:and algebraic topology,
1506:Proc. London Math. Soc.
1492:Theory Appl. Categories
1330:Henle, Michael (1994),
1254:Serre spectral sequence
1140:EilenbergâGanea theorem
657:th homology group of a
616:natural transformations
406:A simplicial 3-complex.
265:. Examples include the
95:is again a free group.
2153:Mathematics portal
2053:Metrics and properties
2039:Second-countable space
1508:, S3-36 (2): 193â212,
1224:Higher category theory
1115:BlakersâMassey theorem
906:Alexander Grothendieck
762:. The main theorem on
694:differential equations
599:differential equations
560:natural transformation
478:combinatorial topology
407:
374:
345:
46:
1407:Topology and Geometry
1219:Grothendieck topology
1101:Gordon Thomas Whyburn
740:-sphere to Euclidean
653:The free rank of the
456:to meet the needs of
405:
375:
346:
318:in three-dimensional
283:real projective plane
56:that uses tools from
40:
27:Branch of mathematics
2108:Invariance of domain
2060:Euler characteristic
2034:Bundle (mathematics)
1769:"Algebraic topology"
1588:Greenberg, Marvin J.
1480:fundamental groupoid
1170:LerayâHirsch theorem
916:Friedrich Hirzebruch
507:, or more generally
466:simplicial complexes
355:
326:
239:of homology theory.
223:algebraic invariants
81:homotopy equivalence
2118:Tychonoff's theorem
2113:Poincaré conjecture
1867:General (point-set)
1234:Homological algebra
1120:BorsukâUlam theorem
1001:Joseph Neisendorfer
734:BorsukâUlam theorem
527:does have a finite
227:algebraic structure
72:topological spaces
2231:Algebraic topology
2103:De Rham cohomology
2024:Polyhedral complex
2014:Simplicial complex
1737:Algebraic topology
1700:van Kampen, Egbert
1682:Algebraic Topology
1659:Algebraic Topology
1619:Algebraic Topology
1467:10.1007/BF01198133
1204:Algebraic K-theory
1108:Important theorems
1096:J. H. C. Whitehead
991:John Coleman Moore
926:Michael J. Hopkins
747:Any subgroup of a
727:hairy ball theorem
682:de Rham cohomology
659:simplicial complex
643:map from the unit
587:de Rham cohomology
572:group homomorphism
525:simplicial complex
505:fundamental groups
454:J. H. C. Whitehead
412:simplicial complex
408:
394:Simplicial complex
370:
341:
308:mathematical knots
119:topological spaces
62:topological spaces
50:Algebraic topology
47:
18:Algebraic Topology
2218:
2217:
2007:fundamental group
1692:978-3-03719-048-7
1555:singular homology
1544:978-3-03719-083-8
1190:Whitehead theorem
1135:DoldâThom theorem
1046:Jean-Pierre Serre
1036:Mikhail Postnikov
966:Saunders Mac Lane
956:Solomon Lefschetz
936:Egbert van Kampen
876:Charles Ehresmann
866:Shiing-Shen Chern
782:; but every such
550:; the notions of
499:(or more general
416:topological space
259:topological space
169:topological space
123:fundamental group
16:(Redirected from
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1933:
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1677:tom Dieck, Tammo
1671:
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1597:
1583:
1552:
1547:, archived from
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1499:
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1445:
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1443:
1434:, archived from
1420:
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1354:
1348:
1346:
1327:
1321:
1319:
1297:Fréchet, Maurice
1293:
1287:
1281:
1155:Hurewicz theorem
1086:Leopold Vietoris
1016:Grigori Perelman
886:Hans Freudenthal
881:Samuel Eilenberg
690:sheaf cohomology
678:smooth manifolds
620:weak equivalence
604:Samuel Eilenberg
595:sheaf cohomology
583:smooth manifolds
464:properties than
379:
377:
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371:
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368:
363:
350:
348:
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306:is the study of
287:Poincaré duality
141:abstract algebra
58:abstract algebra
21:
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2123:Urysohn's lemma
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1897:low-dimensional
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1725:Further reading
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1559:crossed modules
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1402:Bredon, Glen E.
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1160:KĂŒnneth theorem
1110:
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1091:Hassler Whitney
1071:Dennis Sullivan
1066:Norman Steenrod
1031:Nicolae Popescu
946:Hermann KĂŒnneth
931:Witold Hurewicz
851:William Browder
811:
764:covering spaces
717:if and only if
629:
608:Norman Steenrod
579:Georges de Rham
544:
509:homotopy theory
474:
458:homotopy theory
400:
392:Main articles:
390:
382:ambient isotopy
358:
353:
352:
329:
324:
323:
320:Euclidean space
301:
295:
263:Euclidean space
251:
245:
207:cochain complex
205:defined from a
191:homology theory
187:
181:
137:
131:
115:
109:
107:Homotopy groups
101:
52:is a branch of
35:
28:
23:
22:
15:
12:
11:
5:
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2115:
2110:
2105:
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2077:
2072:
2070:Winding number
2067:
2062:
2056:
2054:
2050:
2049:
2047:
2046:
2041:
2036:
2031:
2026:
2021:
2016:
2011:
2010:
2009:
2004:
2002:homotopy group
1994:
1993:
1992:
1987:
1982:
1977:
1972:
1962:
1957:
1952:
1942:
1940:
1936:
1935:
1928:
1926:
1924:
1923:
1918:
1913:
1912:
1911:
1901:
1900:
1899:
1889:
1884:
1879:
1874:
1869:
1863:
1861:
1857:
1856:
1851:
1849:
1848:
1841:
1834:
1826:
1820:
1819:
1783:
1765:
1750:
1732:Hatcher, Allen
1726:
1723:
1722:
1721:
1696:
1691:
1673:
1667:
1654:
1649:
1634:
1628:
1614:Hatcher, Allen
1610:
1604:
1584:
1579:
1571:Addison-Wesley
1562:
1543:
1518:
1501:
1483:
1450:
1422:
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1398:
1383:
1380:
1377:
1376:
1369:
1349:
1342:
1322:
1315:
1288:
1286:, p. 163)
1284:Fraleigh (1976
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1261:
1256:
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1226:
1221:
1216:
1211:
1209:Exact sequence
1206:
1200:
1198:
1195:
1193:
1192:
1187:
1182:
1177:
1172:
1167:
1162:
1157:
1152:
1147:
1142:
1137:
1132:
1127:
1122:
1117:
1111:
1109:
1106:
1104:
1103:
1098:
1093:
1088:
1083:
1078:
1073:
1068:
1063:
1058:
1053:
1051:Isadore Singer
1048:
1043:
1041:Daniel Quillen
1038:
1033:
1028:
1026:Lev Pontryagin
1023:
1021:Henri Poincaré
1018:
1013:
1011:Sergei Novikov
1008:
1003:
998:
993:
988:
983:
978:
973:
968:
963:
958:
953:
948:
943:
938:
933:
928:
923:
918:
913:
908:
903:
901:Israel Gelfand
898:
896:Pierre Gabriel
893:
888:
883:
878:
873:
868:
863:
858:
853:
848:
843:
838:
833:
828:
823:
821:Michael Atiyah
818:
812:
810:
809:Notable people
807:
806:
805:
799:
796:Higgins (1971)
745:
730:
704:
699:A manifold is
697:
674:
651:
628:
625:
614:equipped with
543:
540:
511:, and through
473:
470:
439:simplicial set
389:
386:
367:
362:
338:
333:
297:Main article:
294:
291:
247:Main article:
244:
241:
203:abelian groups
183:Main article:
180:
177:
161:abelian groups
147:(in part from
133:Main article:
130:
127:
113:Homotopy group
111:Main article:
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19:
2210:Publications
2075:Chern number
2065:Betti number
1948: /
1939:Key concepts
1887:Differential
1871:
1811:. Retrieved
1791:
1772:
1736:
1712:(1): 261â7,
1709:
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1566:
1549:the original
1534:math/0407275
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1436:the original
1430:
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1395:
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1359:
1352:
1332:
1325:
1305:
1291:
1279:
1249:Lie groupoid
1081:Hiroshi Toda
1006:Emmy Noether
976:J. Peter May
861:Henri Cartan
856:Ronald Brown
836:Karol Borsuk
831:Armand Borel
826:Enrico Betti
787:
783:
779:
775:
771:
767:
759:
752:
741:
737:
722:
718:
715:vector field
708:
667:Betti number
662:
654:
645:
630:
627:Applications
576:
568:homeomorphic
563:
545:
533:
529:presentation
490:
475:
449:
447:
432:
430:, and their
411:
409:
303:
302:
279:Klein bottle
254:
252:
234:
219:coboundaries
210:
194:
188:
152:
144:
138:
116:
102:
85:
49:
48:
2173:Wikiversity
2090:Key results
1455:Arch. Math.
996:Jack Morava
986:John Milnor
981:Barry Mazur
891:Peter Freyd
816:Frank Adams
462:categorical
304:Knot theory
299:Knot theory
293:Knot theory
54:mathematics
2019:CW complex
1960:Continuity
1950:Closed set
1909:cohomology
1813:2008-09-27
1498:(2): 71â93
1442:2022-08-17
1382:References
961:Jean Leray
941:Daniel Kan
921:Heinz Hopf
841:Raoul Bott
749:free group
701:orientable
641:continuous
564:invariants
548:functorial
521:nonabelian
517:cohomology
482:CW complex
450:CW complex
398:CW complex
273:, and the
229:than does
195:cohomology
185:Cohomology
179:Cohomology
93:free group
66:invariants
2198:geometric
2193:algebraic
2044:Cobordism
1980:Hausdorff
1975:connected
1892:Geometric
1882:Continuum
1872:Algebraic
1779:EMS Press
1475:122228464
1461:: 85â88,
1426:Brown, R.
1076:René Thom
792:groupoids
428:triangles
388:Complexes
312:embedding
243:Manifolds
60:to study
2225:Category
2163:Wikibook
2141:Category
2029:Manifold
1997:Homotopy
1955:Interior
1946:Open set
1904:Homology
1853:Topology
1804:Archived
1789:(1999).
1734:(2002).
1718:51000091
1679:(2008),
1616:(2002),
1428:(2007),
1404:(1993),
1303:(2012),
1239:K-theory
1197:See also
639:: every
612:functors
552:category
513:homology
501:homotopy
255:manifold
249:Manifold
231:homology
215:cocycles
211:cochains
199:sequence
157:sequence
145:homology
135:Homology
129:Homology
89:subgroup
70:classify
2188:general
1990:uniform
1970:compact
1921:Digital
1781:, 2001
1301:Fan, Ky
711:-sphere
661:is the
556:functor
165:modules
2183:Topics
1985:metric
1860:Fields
1787:May JP
1760:
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493:groups
486:groups
420:points
316:circle
271:sphere
269:, the
236:chains
217:, and
1965:Space
1807:(PDF)
1796:(PDF)
1714:JSTOR
1529:arXiv
1471:S2CID
1271:Notes
1259:Sheaf
757:graph
684:, or
648:-disk
602:when
589:, or
414:is a
314:of a
275:torus
267:plane
257:is a
173:group
171:or a
153:homos
151:áœÎŒÏÏ
149:Greek
91:of a
74:up to
68:that
43:torus
1758:ISBN
1756:and
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732:The
706:The
686:Äech
680:via
635:The
606:and
591:Äech
585:via
558:and
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688:or
665:th
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163:or
159:of
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