In
mathematics and specifically in
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, rational homotopy theory is a simplified version of
homotopy theory for
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s, in which all
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
in the
homotopy groups is ignored. It was founded by and . This simplification of homotopy theory makes certain calculations much easier.
Rational homotopy types of
simply connected spaces can be identified with (isomorphism classes of) certain algebraic objects called Sullivan minimal models, which are commutative
differential graded algebras over the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s satisfying certain conditions.
A geometric application was the theorem of Sullivan and Micheline Vigué-Poirrier (1976): every simply connected
closed Riemannian manifold ''X'' whose rational cohomology ring is not generated by one element has infinitely many geometrically distinct
closed geodesic In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flo ...
s. The proof used rational homotopy theory to show that the
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplici ...
s of the
free loop space of ''X'' are unbounded. The theorem then follows from a 1969 result of
Detlef Gromoll
Detlef Gromoll (13 May 1938 – 31 May 2008) was a mathematician who worked in Differential geometry.
Biography
Gromoll was born in Berlin in 1938, and was a classically trained violinist.
After living and attending school in Rosdorf and gra ...
and Wolfgang Meyer.
Rational spaces
A
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
of
simply connected topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s is called a rational homotopy equivalence if it induces an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
on
homotopy groups
tensored with the rational numbers
. Equivalently: ''f'' is a rational homotopy equivalence if and only if it induces an isomorphism on
singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
groups with rational coefficients. The rational homotopy category (of simply connected spaces) is defined to be the
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
of the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of simply connected spaces with respect to rational homotopy equivalences. The goal of rational homotopy theory is to understand this category (i.e. to determine the information that can be recovered from rational homotopy equivalences).
One basic result is that the rational homotopy category is
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
to a
full subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitive ...
of the
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed be ...
of topological spaces, the subcategory of rational spaces. By definition, a rational space is a simply connected
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
all of whose homotopy groups are
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over the rational numbers. For any simply connected CW complex
, there is a rational space
, unique up to
homotopy equivalence
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
, with a map
that induces an isomorphism on homotopy groups tensored with the rational numbers. The space
is called the rationalization of
. This is a special case of Sullivan's construction of the
localization
Localization or localisation may refer to:
Biology
* Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence
* Localization of sensation, ability to tell what part of the body is a ...
of a space at a given set of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s.
One obtains equivalent definitions using homology rather than homotopy groups. Namely, a simply connected CW complex
is a rational space if and only if its homology groups
are rational vector spaces for all
. The rationalization of a simply connected CW complex
is the unique rational space
(up to homotopy equivalence) with a map
that induces an isomorphism on rational homology. Thus, one has
:
and
:
for all
.
These results for simply connected spaces extend with little change to
nilpotent spaces (spaces whose
fundamental group is
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
and acts nilpotently on the higher homotopy groups).
Computing the
homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure o ...
is a central open problem in homotopy theory. However, the ''rational'' homotopy groups of spheres were computed by
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
in 1951:
:
and
:
This suggests the possibility of describing the whole rational homotopy category in a practically computable way. Rational homotopy theory has realized much of that goal.
In homotopy theory,
spheres
The Synchronized Position Hold Engage and Reorient Experimental Satellite (SPHERES) are a series of miniaturized satellites developed by MIT's Space Systems Laboratory for NASA and US Military, to be used as a low-risk, extensible test bed for the ...
and
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
s are two very different types of basic spaces from which all spaces can be built. In rational homotopy theory, these two types of spaces become much closer. In particular, Serre's calculation implies that
is the Eilenberg–MacLane space
. More generally, let ''X'' be any space whose rational cohomology ring is a free
graded-commutative algebra (a
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of a
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
on generators of even degree and an
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
on generators of odd degree). Then the rationalization
is a
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of Eilenberg–MacLane spaces. The hypothesis on the cohomology ring applies to any
compact Lie group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
(or more generally, any
loop space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolo ...
). For example, for the unitary group
SU(''n''),
:
Cohomology ring and homotopy Lie algebra
There are two basic invariants of a space ''X'' in the rational homotopy category: the rational
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
ring
and the homotopy Lie algebra
. The rational cohomology is a graded-commutative algebra over
, and the homotopy groups form a
graded Lie algebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operati ...
via the
Whitehead product. (More precisely, writing
for the loop space of ''X'', we have that
is a graded Lie algebra over
. In view of the isomorphism
, this just amounts to a shift of the grading by 1.) For example, Serre's theorem above says that
is the
free graded Lie algebra on one generator of degree
.
Another way to think of the homotopy Lie algebra is that the homology of the loop space of ''X'' is the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
of the homotopy Lie algebra:
:
Conversely, one can reconstruct the rational homotopy Lie algebra from the homology of the loop space as the subspace of
primitive elements in the
Hopf algebra .
A central result of the theory is that the rational homotopy category can be described in a purely algebraic way; in fact, in two different algebraic ways. First, Quillen showed that the rational homotopy category is equivalent to the homotopy category of connected
differential graded Lie algebra
In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have appl ...
s. (The associated graded Lie algebra
is the homotopy Lie algebra.) Second, Quillen showed that the rational homotopy category is equivalent to the homotopy category of 1-connected differential graded cocommutative
coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
s. (The associated coalgebra is the rational homology of ''X'' as a coalgebra; the
dual vector space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
is the rational cohomology ring.) These equivalences were among the first applications of Quillen's theory of
model categories.
In particular, the second description implies that for any graded-commutative
-algebra ''A'' of the form
:
with each vector space
of finite dimension, there is a simply connected space ''X'' whose rational cohomology ring is isomorphic to ''A''. (By contrast, there are many restrictions, not completely understood, on the integral or mod ''p'' cohomology rings of topological spaces, for prime numbers ''p''.) In the same spirit, Sullivan showed that any graded-commutative
-algebra with
that satisfies
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
is the cohomology ring of some simply connected
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
closed manifold, except in dimension 4''a''; in that case, one also needs to assume that the intersection pairing on
is of the form
over
.
One may ask how to pass between the two algebraic descriptions of the rational homotopy category. In short, a Lie algebra determines a graded-commutative algebra by
Lie algebra cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to p ...
, and an
augmented commutative algebra determines a graded Lie algebra by reduced
André–Quillen cohomology. More generally, there are versions of these constructions for differential graded algebras. This duality between commutative algebras and Lie algebras is a version of
Koszul duality In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohom ...
.
Sullivan algebras
For spaces whose rational homology in each degree has finite dimension, Sullivan classified all rational homotopy types in terms of simpler algebraic objects, Sullivan algebras. By definition, a Sullivan algebra is a commutative differential graded algebra over the rationals
, whose underlying algebra is the free commutative graded algebra
on a graded vector space
:
satisfying the following "nilpotence condition" on its differential ''d'': the space ''V'' is the union of an increasing series of graded subspaces,
, where
on
and
is contained in
. In the context of differential graded algebras ''A'', "commutative" is used to mean graded-commutative; that is,
:
for ''a'' in
and ''b'' in
.
The Sullivan algebra is called minimal if the image of ''d'' is contained in
, where
is the direct sum of the positive-degree subspaces of
.
A Sullivan model for a commutative differential graded algebra ''A'' is a Sullivan algebra
with a homomorphism
which induces an isomorphism on cohomology. If
, then ''A'' has a minimal Sullivan model which is unique up to isomorphism. (Warning: a minimal Sullivan algebra with the same cohomology algebra as ''A'' need not be a minimal Sullivan model for ''A'': it is also necessary that the isomorphism of cohomology be induced by a homomorphism of differential graded algebras. There are examples of non-isomorphic minimal Sullivan models with isomorphic cohomology algebras.)
The Sullivan minimal model of a topological space
For any topological space ''X'', Sullivan defined a commutative differential graded algebra
, called the algebra of
polynomial differential form In algebra, the ring of polynomial differential forms on the standard ''n''-simplex is the differential graded algebra:
:\Omega^*_( = \mathbb _0, ..., t_n, dt_0, ..., dt_n(\sum t_i - 1, \sum dt_i).
Varying ''n'', it determines the simplicial comm ...
s 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
is called a model for the space ''X''. When ''X'' is simply connected, such a model determines the rational homotopy type of ''X''.
To any simply connected CW complex ''X'' with all rational homology groups of finite dimension, there is a minimal Sullivan model
for
, which has the property that
and all the
have finite dimension. This is called the Sullivan minimal model of ''X''; it is unique up to isomorphism. This gives an equivalence between rational homotopy types of such spaces and such algebras, with the properties:
*The rational cohomology of the space is the cohomology of its Sullivan minimal model.
*The spaces of indecomposables in ''V'' are the duals of the rational homotopy groups of the space ''X''.
*The Whitehead product on rational homotopy is the dual of the "quadratic part" of the differential ''d''.
*Two spaces have the same rational homotopy type if and only if their minimal Sullivan algebras are isomorphic.
*There is a simply connected space ''X'' corresponding to each possible Sullivan algebra with
and all the
of finite dimension.
When ''X'' is a smooth manifold, the differential algebra of smooth
differential forms on ''X'' (the
de Rham complex
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly ada ...
) 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''-completed homotopy type of ''X'' for a prime number ''p''. Sullivan's "arithmetic square" reduces many problems in homotopy theory to the combination of rational and ''p''-completed homotopy theory, for all primes ''p''.
The construction of Sullivan minimal models for simply connected spaces extends to nilpotent spaces. For more general fundamental groups, things get more complicated; for example, the rational homotopy groups of a finite CW complex (such as the wedge
) can be infinite-dimensional vector spaces.
Formal spaces
A commutative differential graded algebra ''A'', again with
, 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'' (though it does not have to be the ''minimal'' model). Thus the rational homotopy type of a formal space is completely determined by its cohomology ring.
Examples of formal spaces include spheres,
H-space In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed.
Definition
An H-space consists of a topological space , together wi ...
s,
symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s, and compact
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
s. Formality is preserved under products and
wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if ''X'' and ''Y'' are pointed spaces (i.e. topological spaces with distinguished basepoints x_0 and y_0) the wedge sum of ''X'' and ''Y'' is the qu ...
s. For manifolds, formality is preserved by
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
s.
On the other hand, closed
nilmanifold
In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, th ...
s are almost never formal: if ''M'' is a formal nilmanifold, then ''M'' must be the
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
of some dimension. The simplest example of a non-formal nilmanifold is the Heisenberg manifold, the quotient of the
Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ' ...
of real 3×3 upper triangular matrices with 1's on the diagonal by its subgroup of matrices with integral coefficients. Closed
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
s need not be formal: the simplest example is the Kodaira–Thurston manifold (the product of the Heisenberg manifold with a circle). There are also examples of non-formal, simply connected symplectic closed manifolds.
Non-formality can often be detected by
Massey product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist.
Massey triple product
Le ...
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
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the ...
is a non-formal space: it supports a nontrivial triple Massey product.
Examples
*If ''X'' is a sphere of odd dimension
, its minimal Sullivan model has one generator ''a'' of degree
with
, and a basis of elements 1, ''a''.
*If ''X'' is a sphere of even dimension
, its minimal Sullivan model has two generators ''a'' and ''b'' of degrees
and
, with
,
, and a basis of elements
,
,
, where the arrow indicates the action of ''d''.
*If ''X'' is the
complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
with
, its minimal Sullivan model has two generators ''u'' and ''x'' of degrees 2 and
, with
and
. It has a basis of elements
,
,
.
*Suppose that ''V'' has 4 elements ''a'', ''b'', ''x'', ''y'' of degrees 2, 3, 3 and 4 with differentials
,
,
,
. 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
. Any homomorphism from ''V'' to its cohomology algebra would map ''y'' to 0 and ''x'' to a multiple of ''b''; so it would map
to 0. So ''V'' cannot be a model for its cohomology algebra. The corresponding topological spaces are two spaces with isomorphic rational cohomology rings but different rational homotopy types. Notice that
is in the Massey product
.
Elliptic and hyperbolic spaces
Rational homotopy theory revealed an unexpected dichotomy among finite CW complexes: either the rational homotopy groups are zero in sufficiently high degrees, or they grow
exponentially
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
* Exponential decay, decrease at a rate proportional to value
*Exp ...
. Namely, let ''X'' be a simply connected space such that
is a finite-dimensional
-vector space (for example, a finite CW complex has this property). Define ''X'' to be rationally elliptic if
is also a finite-dimensional
-vector space, and otherwise rationally hyperbolic. Then Félix and Halperin showed: if ''X'' is rationally hyperbolic, then there is a real number
and an integer ''N'' such that
:
for all
.
For example, spheres, complex projective spaces, and
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
s for compact Lie groups are elliptic. On the other hand, "most" finite complexes are hyperbolic. For example:
*The rational cohomology ring of an elliptic space satisfies Poincaré duality.
*If ''X'' is an elliptic space whose top nonzero rational cohomology group is in degree ''n'', then each Betti number
is at most the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
(with equality for the ''n''-dimensional torus).
*The
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
of an elliptic space ''X'' is nonnegative. If the Euler characteristic is positive, then all odd Betti numbers
are zero, and the rational cohomology ring of ''X'' is a
complete intersection ring In commutative algebra, a complete intersection ring is a commutative ring similar to the coordinate rings of varieties that are complete intersections. Informally, they can be thought of roughly as the local rings that can be defined using the "min ...
.
There are many other restrictions on the rational cohomology ring of an elliptic space.
Bott Bott is an English and German surname. Notable people with the surname include:
*Catherine Bott, English soprano
* Charlie Bott, English rugby player
* François Bott (born 1935)
* John Bott
*Leon Bott, Australian rugby league footballer
* Leonida ...
's conjecture predicts that every simply connected closed Riemannian manifold with nonnegative
sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
should be rationally elliptic. Very little is known about the conjecture, although it holds for all known examples of such manifolds.
Halperin's conjecture asserts that the rational
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological ...
of a fiber sequence of simply-connected spaces with rationally elliptic fiber of non-zero Euler characteristic vanishes at the second page.
A simply connected finite complex ''X'' is rationally elliptic if and only if the rational homology of the loop space
grows at most polynomially. More generally, ''X'' is called integrally elliptic if the mod ''p'' homology of
grows at most polynomially, for every prime number ''p''. All known Riemannian manifolds with nonnegative sectional curvature are in fact integrally elliptic.
[Félix, Halperin & Thomas (1993), section 3.]
See also
*
Mandell's theorem – analogue of rational homotopy theory in p-adic settings
*
Chromatic homotopy theory In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups ...
Notes
References
*
*
*
*
*
*
*
*
*
*
*
*
*{{citation, author1-last=Sullivan, author1-first=Dennis, author2-last=Vigué-Poirrier, author2-first=Micheline, title=The homology theory of the closed geodesic problem, year=1976, journal=
Journal of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book ...
, volume=11, issue=4, pages=633–644
, mr=0455028, doi=10.4310/jdg/1214433729
Homotopy theory