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| In mathematics, '''rational homotopy theory''' is the study of the rational homotopy type of a [[topological space|space]], which means roughly that one ignores all [[Torsion (algebra)|torsion]] in the [[homotopy group]]s. It was started by {{harvs|txt=yes|first=Dennis |last=Sullivan|authorlink=Dennis Sullivan|year=1977}} and {{harvs|txt=yes|first=Daniel|last= Quillen|authorlink=Daniel Quillen|year=1969}}.
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| Rational homotopy types of simply connected spaces can be identified with (isomorphism classes of) certain algebraic objects called minimal Sullivan algebras,
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| which are [[Commutativity|commutative]] [[differential graded algebra]]s over the [[rational number]]s satisfying certain conditions.
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| The standard textbook on rational homotopy theory is {{harv|Félix|Halperin|Thomas|2001}}.
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| ==Rational spaces==
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| A '''rational space''' is a [[simply connected space]] all of whose homotopy groups are [[vector space]]s over the rational numbers. If ''X'' is any simply connected [[CW complex]], then there is a rational space ''Y'', unique up to [[homotopy equivalence]], and a map from ''X'' to ''Y'' inducing an [[isomorphism]] on homotopy groups [[tensor product|tensored]] with the rational numbers. The space ''Y'' is called the '''rationalization''' of ''X'', and is the [[localization of a topological space|localization]] of ''X'' at the rationals, and is the '''rational homotopy type''' of ''X''. Informally, it is obtained from ''X'' by killing all torsion in the homotopy groups of ''X''.
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| ==Sullivan algebras==
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| A '''Sullivan algebra''' is a commutative differential graded algebra over the rationals '''Q''', whose underlying algebra is the free commutative graded algebra Λ(''V'') on a graded vector space
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| :<math>V=\oplus_{n>0}V^n,\,</math>
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| satisfying the following "nilpotence condition on ''d'' ": ''V'' is the union of an increasing series of graded subspaces ''V''(0)⊆''V''(1)⊆
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| where ''d'' = 0 on ''V''(0) and ''d''(''V''(''k'')) is contained in Λ(''V''(''k'' − 1)). Here "commutative" means commutative in the graded sense, sometimes called [[supercommutative]]. Thus ''ab'' = (−1)<sup>deg(''a'')deg(''b'')</sub>''ba''.)
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| The Sullivan algebra is called '''minimal''' if the image of ''d'' is contained in Λ<sup>+</sup>(''V'')<sup>2</sup>, where Λ<sup>+</sup>(''V'') is the direct sum of the positive degree subspaces of Λ(''V'').
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| A '''Sullivan model''' for a commutative differential graded algebra ''A'' is an [[algebra homomorphism]] from a Sullivan algebra Λ(''V'') that is an isomorphism on [[cohomology]]. If ''A''<sup>0</sup> = '''Q''' then ''A'' has a minimal Sullivan model which is unique up to isomorphism. (Warning: a minimal Sullivan algebra with the same cohomology as ''A'' need not be a minimal Sullivan model for ''A'': it is also necessary that the isomorphism of cohomology be induced by an algebra homomorphism. There are examples of non-isomorphic minimal Sullivan models with the same cohomology algebra.)
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| ==The Sullivan minimal model of a topological space==
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| For any topological space ''X'' Sullivan defined a commutative differential graded algebra ''A''<sub>''PL''</sub>(''X''), called the algebra of polynomial differential forms on ''X'' with rational coefficients. An element of this algebra consists of (roughly) a polynomial form on each singular simplex of ''X'', compatible with face and degeneracy maps. This algebra is usually very large (uncountable dimension) but can be replaced by a much smaller algebra. More precisely, any differential graded algebra with the same Sullivan minimal model as ''A''<sub>''PL''</sub>(''X'') is called a '''model''' for the space ''X'', and determines the rational homotopy type of ''X'' when ''X'' is simply connected.
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| To any simply connected CW complex ''X'' with all rational homology groups of finite dimension one can assign a minimal Sullivan algebra Λ''V'' of ''A''<sub>''PL''</sub>(''X''), which has the property that ''V''<sup>1</sup> = 0 and all the ''V''<sup>''k''</sup> of finite dimension. This is called the Sullivan minimal model of ''X'', and is unique up to isomorphism. This gives an equivalence between rational homotopy types of such spaces and such algebras, such that:
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| *The rational cohomology of the space is the cohomology of its Sullivan minimal model.
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| *The spaces of indecomposables in ''V'' are the duals of the rational homotopy groups of the space ''X''.
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| *The Whitehead product on rational homotopy is the dual of the "quadratic part" of the differential ''d''.
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| *Two spaces have the same rational homotopy type if and only if their minimal Sullivan algebras are isomorphic.
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| *There is a simply connected space ''X'' corresponding to each possible Sullivan algebra with ''V''<sup>1</sup> = 0 and all the ''V''<sup>''k''</sup> of finite dimension.
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| When ''X'' is a [[smooth manifold]], the differential algebra of smooth [[differential form]]s on ''X'' (the [[De Rham cohomology|de Rham complex]]) is almost a model for ''X''; more precisely it is the tensor product of a model for ''X'' with the reals and therefore determines the '''real homotopy type'''. One can go further and define the '''p-adic homotopy type''' and the '''adelic homotopy type''' and compare them to the rational homotopy type.
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| The results above for simply connected spaces can easily be extended to nilpotent spaces (whose fundamental group is [[nilpotent group|nilpotent]] and acts nilpotently on the higher homotopy groups). For more general fundamental groups things get more complicated; for example, the homotopy groups need not be finitely generated even if there are only a finite number of cells of the CW complex in each dimension. | |
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| ==Formal spaces==
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| A commutative differential graded algebra ''A'', again with ''A''<sup>0</sup> = '''Q''', is called '''formal''' if ''A'' has a model with vanishing differential. This is equivalent to requiring that the cohomology algebra of ''A'' (viewed as a differential algebra with trivial differential) is a model for ''A''. Thus two formal commutative differential graded algebras with the isomorphic cohomology algebras have the same Sullivan minimal model. A space is called formal if its minimal Sullivan model is formal, so the minimal Sullivan model of a simply connected formal topological space is determined by the rational cohomology ring. This means that the rational homotopy of a formal space is particularly easy to work out.
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| Examples of formal spaces include [[Hypersphere|spheres]], [[H-space]]s, [[symmetric space]]s, and [[compact space|compact]] [[Kähler manifold]]s {{harv|Deligne|Griffiths|Morgan|Sullivan|1989}}. Formality is preserved under [[wedge sum]]s and [[direct product]]s; it is also preserved under [[connected sum]]s for manifolds.
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| On the other hand, [[nilmanifold]]s are almost never formal: if ''M<sup>n</sup>'' is a compact formal nilmanifold, then ''M<sup>n</sup>=T<sup>n</sup>'', the ''n''-dimensional [[torus]] {{harv|Hasegawa|1975}}. The simplest example of a non-formal compact nilmanifold is the '''Heisenberg manifold''', the quotient of the [[Heisenberg group]] of 3×3 upper triangular matrices with 1's on the diagonal by its subgroup of matrices with integral coefficients. [[Symplectic manifold]]s need not be formal: the simplest example is the Kodaira-Thurston manifold (the product of the Heisenberg manifold with a circle). Examples of non-formal, simply connected symplectic manifolds were given in {{harvtxt|Babenko|Taimanov|2000}}.
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| Non-formality may often be detected by [[Massey product]]s. Indeed, if a differential graded algebra ''A'' is formal, then all (higher order) Massey products must vanish. The converse is not true: formality means, roughly speaking, the "uniform" vanishing of all Massey products. The complement of the [[Borromean rings]] is a non-formal space: it supports a non-trivial triple Massey product.
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| {{harvtxt|Halperin|Stasheff|1979}} gave an algorithm for deciding whether or not a commutative differential graded algebra is formal.
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| ==Examples==
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| *If ''X'' is a sphere of odd dimension 2''n'' + 1 > 1, its minimal Sullivan model has 1 generator ''a'' of degree 2''n'' + 1 with ''da'' = 0, and a basis of elements 1, ''a''.
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| *If ''X'' is a sphere of even dimension 2''n'' > 0, its minimal Sullivan model has 2 generators ''a'' and ''b'' of degrees 2''n'' and 4''n'' − 1, with ''db'' = ''a''<sup>2</sup>, ''da'' = 0, and a basis of elements 1, ''a'', ''b''→ ''a''<sup>2</sup>, ''ab''→''a''<sup>3</sup>, ''a''<sup>2</sup>''b''→''a''<sup>4</sup>, ... where the arrow indicated the action of ''d''.
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| *Suppose that ''V'' has 4 elements ''a'', ''b'', ''x'', ''y'' of degrees 2, 3, 3 and 4 with differentials ''da'' = 0, ''db'' = 0, ''dx'' = ''a''<sup>2</sup>, ''dy'' = ''ab''. Then this algebra is a minimal Sullivan algebra that is not formal. The cohomology algebra has nontrivial components only in dimension 2,3,6, generated respectively by ''a'', ''b'' and ''xb-ay''. Any homomorphism from ''V'' to its cohomology algebra would map ''y'' to 0, ''x'' to a multiple of ''b'', so it would surely map ''xb-ay'' to 0. So ''V'' cannot be a model for its cohomology algebra. The corresponding topological spaces are two spaces with the same rational [[cohomology ring]] but different rational homotopy types. Notice that ''xb-ay'' is in the Massey product <math> \langle [a], [a], [b] \rangle </math>.
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| ==External links==
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| *[http://www.math.uic.edu/~bshipley/hess_ratlhtpy.pdf Rational Homotopy Theory: A Brief Introduction] by Kathryn Hess
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| ==References==
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| *{{citation|last=Babenko|first= Ivan K.|last2= Taimanov |first2=Iskander A.|title=On nonformal simply connected symplectic manifolds|journal=Siberian Mathematical Journal |volume=41 |year=2000|issue= 2|pages=204–217|url=http://www.springerlink.com/content/h3674t8t5635l802/|doi=10.1007/BF02674589|mr=1762178}}
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| *{{citation|last=Deligne|first=Pierre |last2=Griffiths|first2= Phillip A.|last3= Morgan|first3= John W.|last4=Sullivan|first4=Dennis|title=Real homotopy theory of Kähler manifolds|journal=[[Inventiones Mathematicae]]|volume=29|year=1975|pages=245–274|url=http://www.springerlink.com/content/m48544t785221635/|doi=10.1007/BF01389853|mr=0382702|issue=3 }}
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| *{{citation|last= Félix|first= Yves|last2= Halperin|first2= Stephen|last3= Thomas|first3= Jean-Claude |title=Rational homotopy theory|series= Graduate Texts in Mathematics|volume= 205|publisher= Springer-Verlag|publication-place= New York|year= 2001|pages= xxxiv+535 | isbn= 0-387-95068-0|mr= 1802847}}
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| *{{citation|last=Griffiths|first= Phillip A.|last2= Morgan|first2= John W.|title=Rational homotopy theory and differential forms|series=Progress in Mathematics|volume= 16|publisher= Birkhäuser|year= 1981 |pages=xi+242 pp. |isbn= 3-7643-3041-4|publication-place= Boston, Mass.|mr=0641551 }}
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| *{{citation|last=Halperin|first=Stephen |last2=Stasheff|first2= James|author2-link=Jim Stasheff|title=Obstructions to homotopy equivalences|journal=[[Advances in Mathematics]]|volume=32|year=1979|pages=233–279|doi=10.1016/0001-8708(79)90043-4|mr=0539532|issue=3 }}
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| *{{citation|last=Hasegawa|first=Keizo|title=Minimal models of nilmanifolds|journal=[[Proceedings of the American Mathematical Society]]|volume=106|year=1989|pages=65–71|doi=10.2307/2047375|issue=1|publisher=Proceedings of the American Mathematical Society, Vol. 106, No. 1|mr=946638|jstor=2047375 }}
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| *{{citation|last=Hess|first= Kathryn|chapter=A history of rational homotopy theory|title= History of topology|pages= 757–796|publisher= North-Holland |publication-place=Amsterdam|year= 1999|editor-first=I. M.|editor-last= James| isbn=0-444-82375-1|mr=1721122 }}
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| *{{citation|first=D. |last=Quillen|title=Rational homotopy theory|journal=[[Annals of Mathematics]]|volume=90|year=1969|pages=205–295
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| |doi=10.2307/1970725|issue=2|publisher=The Annals of Mathematics, Vol. 90, No. 2 |jstor=1970725|mr=0258031}}
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| *{{citation| last=Sullivan|first=Dennis|title=Infinitesimal computations in topology|year=1977|journal=[[Publications Mathématiques de l'IHÉS]]|volume=47|pages=269–331
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| |url=http://www.numdam.org/item?id=PMIHES_1977__47__269_0| mr=0646078}}
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| *{{springer|title=Rational homotopy theory|id=R/r077600|last=Sullivan|first=Dennis}}
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| [[Category:Homotopy theory]]
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Wilber Berryhill is what his wife enjoys to contact him and he totally enjoys this title. I am currently a journey agent. The preferred hobby for him and his children is to perform lacross and he would never give it up. Alaska is the only location I've been residing in but now I'm contemplating other options.
My site ... real psychic (modenpeople.co.kr)