Difference between revisions of "Sullivan minimal model"
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| − | + | The theory of minimal models began with the work of D. Quillen {{Cite|Qu}}. A simply-connected [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s1203101.png" /> (cf. also [[Simply-connected domain|Simply-connected domain]]) is called rational if its homotopy groups are rational vector spaces (cf. also [[Homotopy group|Homotopy group]]; [[Vector space|Vector space]]). The rationalization functor associates to each simply-connected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s1203102.png" /> a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s1203103.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s1203104.png" /> is rational and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s1203105.png" /> is an isomorphism. The interest of this construction is that the homotopy category of rational spaces has an algebraic nature. More precisely, in {{Cite|Qu}}, D. Quillen established an equivalence of homotopy categories between the homotopy category of simply-connected rational spaces and the homotopy category of connected differential graded Lie algebras (cf. also [[Lie algebra, graded|Lie algebra, graded]]). | |
| + | |||
| + | In {{Cite|Su}}, D. Sullivan associated to each space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s1203106.png" /> a commutative differential [[Graded algebra|graded algebra]] (CDGA), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s1203107.png" />, which is linked to the cochain algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s1203108.png" /> by a chain of differential graded algebra quasi-isomorphisms (i.e. morphisms inducing isomorphisms in cohomology). This, in particular, gave a solution to Thom's problem of constructing commutative cochains over the rationals. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s1203109.png" />-functor together with its adjoint, the realization functor of a commutative differential graded algebra, induce an equivalence of homotopy categories between the homotopy category of simply-connected rational spaces with finite Betti numbers and the homotopy category of rational commutative differential graded algebras, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031010.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031013.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031014.png" />. | ||
The correspondence | The correspondence | ||
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031034.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031034.png" /></td> </tr></table> | ||
| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031036.png" /> are quasi-isomorphisms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031038.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031040.png" /> are the canonical injection and projection. In this case, the Grivel–Halperin–Thomas theorem asserts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031041.png" /> is a Sullivan minimal model for the homotopy fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031042.png" /> | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031036.png" /> are quasi-isomorphisms, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031038.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031040.png" /> are the canonical injection and projection. In this case, the Grivel–Halperin–Thomas theorem asserts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031041.png" /> is a Sullivan minimal model for the homotopy fibre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031042.png" /> {{Cite|Ha2}}. |
| − | A key result in the theory is the so-called mapping theorem | + | A key result in the theory is the so-called mapping theorem {{Cite|FéHa}}. Recall that the Lyusternik–Shnirel'man category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031043.png" /> is the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031045.png" /> can be covered by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031046.png" /> open sets each contractible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031047.png" /> (cf. also [[Category (in the sense of Lyusternik–Shnirel'man)|Category (in the sense of Lyusternik–Shnirel'man)]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031048.png" /> is a mapping between simply-connected spaces and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031049.png" /> is injective, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031050.png" />. The Lyusternik–Shnirel'man category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031051.png" /> can be computed directly from its Sullivan minimal model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031052.png" />. Indeed, consider the following commutative diagram: |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031053.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031053.png" /></td> </tr></table> | ||
| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031055.png" /> denote the canonical projection and injection and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031056.png" /> is a quasi-isomorphism. The category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031057.png" /> is then the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031058.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031059.png" /> admits a [[Retraction|retraction]] | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031055.png" /> denote the canonical projection and injection and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031056.png" /> is a quasi-isomorphism. The category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031057.png" /> is then the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031058.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031059.png" /> admits a [[Retraction|retraction]] {{Cite|FéHa}}. |
| − | To obtain properties of simply-connected spaces with finite category, it is therefore sufficient to consider Sullivan minimal models <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031060.png" /> with finite category. Using this procedure, Y. Félix, S. Halperin and J.-C. Thomas have obtained the following dichotomy theorem: Either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031061.png" /> is finite-dimensional (the space is called elliptic), or else the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031062.png" /> has exponential growth (the space is thus called hyperbolic) | + | To obtain properties of simply-connected spaces with finite category, it is therefore sufficient to consider Sullivan minimal models <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031060.png" /> with finite category. Using this procedure, Y. Félix, S. Halperin and J.-C. Thomas have obtained the following dichotomy theorem: Either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031061.png" /> is finite-dimensional (the space is called elliptic), or else the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031062.png" /> has exponential growth (the space is thus called hyperbolic) {{Cite|FéHaTh}}. |
| − | When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031063.png" /> is elliptic, the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031064.png" /> is finite, the [[Euler characteristic|Euler characteristic]] is non-negative and the rational cohomology algebra satisfies [[Poincaré duality|Poincaré duality]] | + | When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031063.png" /> is elliptic, the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031064.png" /> is finite, the [[Euler characteristic|Euler characteristic]] is non-negative and the rational cohomology algebra satisfies [[Poincaré duality|Poincaré duality]] {{Cite|Ha}}. |
The minimal model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031065.png" /> contains all the rational homotopy invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031066.png" />. For instance, the cochain algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031067.png" /> is a model for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031068.png" />th Postnikov tower <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031069.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031070.png" /> (cf. also [[Postnikov system|Postnikov system]]), and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031071.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031072.png" /> is the dual of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031073.png" />st <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031074.png" />-invariant | The minimal model of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031065.png" /> contains all the rational homotopy invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031066.png" />. For instance, the cochain algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031067.png" /> is a model for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031068.png" />th Postnikov tower <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031069.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031070.png" /> (cf. also [[Postnikov system|Postnikov system]]), and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031071.png" /> induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031072.png" /> is the dual of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031073.png" />st <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120310/s12031074.png" />-invariant | ||
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====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|FéHa}}||valign="top"| Y. Félix, S. Halperin, "Rational LS category and its applications" ''Trans. Amer. Math. Soc.'', '''273''' (1982) pp. 1–37 {{MR|0664027}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|FéHaTh}}||valign="top"| Y. Félix, S. Halperin, J.C. Thomas, "Rational homotopy theory" (in preparation) {{MR|1802847}} {{ZBL|0961.55002}} {{ZBL|0691.55001}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ha}}||valign="top"| S. Halperin, "Finiteness in the minimal models of Sullivan" ''Trans. Amer. Math. Soc.'', '''230''' (1977) pp. 173–199 {{MR|0461508}} {{ZBL|0364.55014}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Ha2}}||valign="top"| S. Halperin, "Lectures on minimal models" ''Mémoire de la SMF'', '''9/10''' (1983) {{MR|0736299}} {{MR|0637558}} {{ZBL|0536.55003}} {{ZBL|0505.55014}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Qu}}||valign="top"| D. Quillen, "Rational homotopy theory" ''Ann. of Math.'', '''90''' (1969) pp. 205–295 {{MR|0258031}} {{ZBL|0191.53702}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|Su}}||valign="top"| D. Sullivan, "Infinitesimal computations in topology" ''Publ. IHES'', '''47''' (1977) pp. 269–331 {{MR|0646078}} {{ZBL|0374.57002}} | ||
| + | |- | ||
| + | |} | ||
Revision as of 23:50, 26 February 2012
2020 Mathematics Subject Classification: Primary: 57D99 Secondary: 55D9958A10 [MSN][ZBL]
The theory of minimal models began with the work of D. Quillen [Qu]. A simply-connected topological space 
(cf. also Simply-connected domain) is called rational if its homotopy groups are rational vector spaces (cf. also Homotopy group; Vector space). The rationalization functor associates to each simply-connected space 
a mapping 
, such that 
is rational and 
is an isomorphism. The interest of this construction is that the homotopy category of rational spaces has an algebraic nature. More precisely, in [Qu], D. Quillen established an equivalence of homotopy categories between the homotopy category of simply-connected rational spaces and the homotopy category of connected differential graded Lie algebras (cf. also Lie algebra, graded).
In [Su], D. Sullivan associated to each space 
a commutative differential graded algebra (CDGA), 
, which is linked to the cochain algebra 
by a chain of differential graded algebra quasi-isomorphisms (i.e. morphisms inducing isomorphisms in cohomology). This, in particular, gave a solution to Thom's problem of constructing commutative cochains over the rationals. The 
-functor together with its adjoint, the realization functor of a commutative differential graded algebra, induce an equivalence of homotopy categories between the homotopy category of simply-connected rational spaces with finite Betti numbers and the homotopy category of rational commutative differential graded algebras, 
, such that 
, 
, and 
for each 
.
The correspondence
 |
 |
 |
behaves well with respect to fibrations and cofibrations (cf. also Fibration). Rational homotopy invariants of a space are most easily obtained by means of constructions in the category of commutative differential graded algebras. This procedure has been made very powerful with the Sullivan minimal models.
Let 
be a commutative differential graded algebra such that 
, 
, and 
for each 
. There exists then a quasi-isomorphism of commutative differential graded algebras 
, where 
denotes the free commutative algebra on the graded vector space of finite type 
, and 
. The cochain algebra 
is called the Sullivan minimal model of 
; it is unique up to isomorphism.
The Sullivan minimal model of 
is called the Sullivan minimal model of 
. It satisfies 
and 
. More generally, for each continuous mapping 
, there is a commutative diagram
 |
where 
and 
are quasi-isomorphisms, 
, 
, and where 
and 
are the canonical injection and projection. In this case, the Grivel–Halperin–Thomas theorem asserts that 
is a Sullivan minimal model for the homotopy fibre of 
[Ha2].
A key result in the theory is the so-called mapping theorem [FéHa]. Recall that the Lyusternik–Shnirel'man category of 
is the least integer 
such that 
can be covered by 
open sets each contractible in 
(cf. also Category (in the sense of Lyusternik–Shnirel'man)). If 
is a mapping between simply-connected spaces and if 
is injective, then 
. The Lyusternik–Shnirel'man category of 
can be computed directly from its Sullivan minimal model 
. Indeed, consider the following commutative diagram:
 |
where 
and 
denote the canonical projection and injection and 
is a quasi-isomorphism. The category of 
is then the least integer 
such that 
admits a retraction [FéHa].
To obtain properties of simply-connected spaces with finite category, it is therefore sufficient to consider Sullivan minimal models 
with finite category. Using this procedure, Y. Félix, S. Halperin and J.-C. Thomas have obtained the following dichotomy theorem: Either 
is finite-dimensional (the space is called elliptic), or else the sequence 
has exponential growth (the space is thus called hyperbolic) [FéHaTh].
When 
is elliptic, the dimension of 
is finite, the Euler characteristic is non-negative and the rational cohomology algebra satisfies Poincaré duality [Ha].
The minimal model of 
contains all the rational homotopy invariants of 
. For instance, the cochain algebra 
is a model for the 
th Postnikov tower 
of 
(cf. also Postnikov system), and the mapping 
induced by 
is the dual of the 
st 
-invariant
 |
 |
The quadratic part of the differential 
is dual to the Whitehead product in 
. More precisely, 
, 
, 
, 
.
References
| [FéHa] | Y. Félix, S. Halperin, "Rational LS category and its applications" Trans. Amer. Math. Soc., 273 (1982) pp. 1–37 MR0664027 |
| [FéHaTh] | Y. Félix, S. Halperin, J.C. Thomas, "Rational homotopy theory" (in preparation) MR1802847 Zbl 0961.55002 Zbl 0691.55001 |
| [Ha] | S. Halperin, "Finiteness in the minimal models of Sullivan" Trans. Amer. Math. Soc., 230 (1977) pp. 173–199 MR0461508 Zbl 0364.55014 |
| [Ha2] | S. Halperin, "Lectures on minimal models" Mémoire de la SMF, 9/10 (1983) MR0736299 MR0637558 Zbl 0536.55003 Zbl 0505.55014 |
| [Qu] | D. Quillen, "Rational homotopy theory" Ann. of Math., 90 (1969) pp. 205–295 MR0258031 Zbl 0191.53702 |
| [Su] | D. Sullivan, "Infinitesimal computations in topology" Publ. IHES, 47 (1977) pp. 269–331 MR0646078 Zbl 0374.57002 |
How to Cite This Entry:
Sullivan minimal model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sullivan_minimal_model&oldid=11595
Sullivan minimal model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sullivan_minimal_model&oldid=11595
This article was adapted from an original article by Yves Félix (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article