Homology sphere
Encyclopedia
In algebraic topology
, a homology sphere is an n-manifold
X having the homology groups of an n-sphere
, for some integer n ≥ 1. That is,
and
Therefore X is a connected space
, with one non-zero higher Betti number
: bn. It does not follow that X is simply connected, only that its fundamental group
is perfect
(see Hurewicz theorem
).
A rational homology sphere is defined similarly but using homology with rational coefficients.
homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a spherical 3-manifold
, it is the only homology 3-sphere (besides the 3-sphere
itself) with a finite fundamental group
. Its fundamental group is known as the binary icosahedral group and has order 120. This shows the Poincaré conjecture
cannot be stated in homology terms alone.
each pair of opposite faces together using this identification yields a closed 3-manifold. (See Seifert–Weber space for a similar construction, using more "twist", that results in a hyperbolic 3-manifold
.)
Alternatively, the Poincaré homology sphere can be constructed as the quotient space
SO(3)/I where I is the icosahedral group
(i.e. the rotational symmetry group
of the regular icosahedron
and dodecahedron, isomorphic to the alternating group A5). More intuitively, this means that the Poincaré homology sphere is the space of all geometrically distinguishable positions of an icosahedron (with fixed center and diameter) in Euclidean 3-space. One can also pass instead to the universal cover of SO(3) which can be realized as the group of unit quaternion
s and is homeomorphic to the 3-sphere. In this case, the Poincaré homology sphere is isomorphic to S3/Ĩ where Ĩ is the binary icosahedral group, the perfect double cover of I embedded
in S3.
Another approach is by Dehn surgery
. The Poincaré homology sphere results from +1 surgery on the right-handed trefoil knot
.
spacecraft led to the suggestion, by Jean-Pierre Luminet
of the Observatoire de Paris and colleagues, that the shape of the Universe
is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft.
There is as yet no strong support for the correctness of the model, however.
on the choice of b′s.) If r is at most 2 this is just the usual 3-sphere; otherwise they are distinct non-trivial homology spheres. If the a′s are 2, 3, and 5 this gives the Poincaré sphere. If there are at least 3 a′s, not 2, 3, 5, then this is an acyclic homology 3-sphere with infinite fundamental group that has a Thurston geometry modeled on the universal cover of SL2(R).
that is not a topological manifold
. The double suspension of A is homeomorphic to the standard 5-sphere, but its triangulation
(induced by some triangulation of A) is not a PL manifold. In other words, this gives an example of a finite simplicial complex
that is a topological manifold but not a PL manifold. (It is not a PL manifold because the link of a point is not always a 4-sphere.)
Galewski and Stern showed that all compact topological manifolds (without boundary) of dimension at least 5 are homeomorphic to simplicial complexes if and only if
there is a homology 3 sphere Σ with Rokhlin invariant 1 such that the connected sum Σ#Σ of Σ with itself bounds a smooth acyclic 4-manifold. the existence of such a homology 3-sphere was an unsolved problem.
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, a homology sphere is an n-manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
X having the homology groups of an n-sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, for some integer n ≥ 1. That is,
- H0(X,Z) = Z = Hn(X,Z)
and
- Hi(X,Z) = {0} for all other i.
Therefore X is a connected space
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, with one non-zero higher Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
: bn. It does not follow that X is simply connected, only that its fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
is perfect
Perfect group
In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
(see Hurewicz theorem
Hurewicz theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism...
).
A rational homology sphere is defined similarly but using homology with rational coefficients.
Poincaré homology sphere
The PoincaréHenri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere. Being a spherical 3-manifold
Spherical 3-manifold
In mathematics, a spherical 3-manifold M is a 3-manifold of the formM=S^3/\Gammawhere \Gamma is a finite subgroup of SO acting freely by rotations on the 3-sphere S^3. All such manifolds are prime, orientable, and closed...
, it is the only homology 3-sphere (besides the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
itself) with a finite fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
. Its fundamental group is known as the binary icosahedral group and has order 120. This shows the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
cannot be stated in homology terms alone.
Construction
A simple construction of this space begins with a dodecahedron. Each face of the dodecahedron is identified with its opposite face, using the minimal clockwise twist to line up the faces. GluingQuotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
each pair of opposite faces together using this identification yields a closed 3-manifold. (See Seifert–Weber space for a similar construction, using more "twist", that results in a hyperbolic 3-manifold
Hyperbolic 3-manifold
A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...
.)
Alternatively, the Poincaré homology sphere can be constructed as the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
SO(3)/I where I is the icosahedral group
Icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...
(i.e. the rotational symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
of the regular icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
and dodecahedron, isomorphic to the alternating group A5). More intuitively, this means that the Poincaré homology sphere is the space of all geometrically distinguishable positions of an icosahedron (with fixed center and diameter) in Euclidean 3-space. One can also pass instead to the universal cover of SO(3) which can be realized as the group of unit quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s and is homeomorphic to the 3-sphere. In this case, the Poincaré homology sphere is isomorphic to S3/Ĩ where Ĩ is the binary icosahedral group, the perfect double cover of I embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in S3.
Another approach is by Dehn surgery
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
. The Poincaré homology sphere results from +1 surgery on the right-handed trefoil knot
Trefoil knot
In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...
.
Cosmology
In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAPWilkinson Microwave Anisotropy Probe
The Wilkinson Microwave Anisotropy Probe — also known as the Microwave Anisotropy Probe , and Explorer 80 — is a spacecraft which measures differences in the temperature of the Big Bang's remnant radiant heat — the Cosmic Microwave Background Radiation — across the full sky. Headed by Professor...
spacecraft led to the suggestion, by Jean-Pierre Luminet
Jean-Pierre Luminet
Jean-Pierre Luminet is a French astrophysicist, specialized in black holes and cosmology. He works as research director for the CNRS , and is a member of the Laboratoire Univers et Théories of the observatory of Paris-Meudon.The asteroid 5523 Luminet, was named after him .-Timeline:* 2003 - An...
of the Observatoire de Paris and colleagues, that the shape of the Universe
Shape of the Universe
The shape of the universe is a matter of debate in physical cosmology over the local and global geometry of the universe which considers both curvature and topology, though, strictly speaking, it goes beyond both...
is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft.
There is as yet no strong support for the correctness of the model, however.
Constructions and examples
- Surgery on a knot in the 3-sphere S3 with framing +1 or − 1 gives a homology sphere.
- More generally, surgery on a link gives a homology sphere whenever the matrix given by intersection numbers (off the diagonal) and framings (on the diagonal) has determinant +1 or −1.
- If p, q, and r are pairwise relatively prime positive integers then the link of the singularity xp + yq + zr = 0 (in other words, the intersection of a small 5-sphere around 0 with this complex surface) is a homology 3-sphere, called a Brieskorn 3-sphere Σ(p, q, r). It is homeomorphic to the standard 3-sphere if one of p, q, and r is 1, and Σ(2, 3, 5) is the Poincaré sphere.
- The connected sum of two oriented homology 3-spheres is a homology 3-sphere. A homology 3-sphere that cannot be written as a connected sum of two homology 3-spheres is called irreducible or prime, and every homology 3-sphere can be written as a connected sum of prime homology 3-spheres in an essentially unique way. (See Prime decomposition (3-manifold)Prime decomposition (3-manifold)In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique collection of prime 3-manifolds....
.)
- Suppose that a1, ..., ar are integers all at least 2 such that any two are coprime. Then the Seifert fiber spaceSeifert fiber spaceA Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a S^1-bundle over a 2-dimensional orbifold...
- over the sphere with exceptional fibers of degrees a1, ..., ar is a homology sphere, where the bs are chosen so that
on the choice of b′s.) If r is at most 2 this is just the usual 3-sphere; otherwise they are distinct non-trivial homology spheres. If the a′s are 2, 3, and 5 this gives the Poincaré sphere. If there are at least 3 a′s, not 2, 3, 5, then this is an acyclic homology 3-sphere with infinite fundamental group that has a Thurston geometry modeled on the universal cover of SL2(R).
Invariants
- The Rokhlin invariant is a Z/2Z valued invariant of homology 3-spheres.
- The Casson invariantCasson invariantIn 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson....
is an integer valued invariant of homology 3-spheres, whose reduction mod 2 is the Rokhlin invariant.
Applications
If A is a homology 3-sphere not homeomorphic to the standard 3-sphere, then the suspension of A is an example of a 4-dimensional homology manifoldHomology manifold
In mathematics, a homology manifold is a locally compact topological space X that looks locally like a topological manifold from the point of view of homology theory.-Definition:...
that is not a topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
. The double suspension of A is homeomorphic to the standard 5-sphere, but its triangulation
Triangulation (topology)
In mathematics, topology generalizes the notion of triangulation in a natural way as follows:A triangulation of a topological space X is a simplicial complex K, homeomorphic to X, together with a homeomorphism h:K\to X....
(induced by some triangulation of A) is not a PL manifold. In other words, this gives an example of a finite simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
that is a topological manifold but not a PL manifold. (It is not a PL manifold because the link of a point is not always a 4-sphere.)
Galewski and Stern showed that all compact topological manifolds (without boundary) of dimension at least 5 are homeomorphic to simplicial complexes if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
there is a homology 3 sphere Σ with Rokhlin invariant 1 such that the connected sum Σ#Σ of Σ with itself bounds a smooth acyclic 4-manifold. the existence of such a homology 3-sphere was an unsolved problem.
Selected reading
- Emmanuel Dror, Homology spheres, Israel Journal of Mathematics 15 (1973), 115–129.
- David Galewski, Ronald Stern Classification of simplicial triangulations of topological manifolds, Annals of MathematicsAnnals of MathematicsThe Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
111 (1980), no. 1, pp. 1–34. - Robion KirbyRobion KirbyRobion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology...
, Martin Scharlemann, Eight faces of the Poincaré homology 3-sphere. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 113–146, Academic PressAcademic PressAcademic Press is an academic book publisher. Originally independent, it was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier....
, New York-London, 1979. - Michel KervaireMichel KervaireMichel André Kervaire was a French mathematician who made significant contributions to topology and algebra. He was the first to show the existence of topological n-manifolds with no differentiable structure , and computed the number of exotic spheres in dimensions greater than four...
, Smooth homology spheres and their fundamental groups, Transactions of the American Mathematical SocietyTransactions of the American Mathematical SocietyTransactions of the American Mathematical Society is a monthly mathematics journal published by the American Mathematical Society. It started in 1900...
144 (1969) 67–72. - Nikolai Saveliev, Invariants of Homology 3-Spheres, Encyclopaedia of Mathematical Sciences, vol 140. Low-Dimensional Topology, I. Springer-Verlag, Berlin, 2002. ISBN 3-540-43796-7
External links
- A 16-Vertex Triangulation of the Poincaré Homology 3-Sphere and Non-PL Spheres with Few Vertices by Anders BjörnerAnders BjörnerAnders Björner is a Swedish professor of mathematics, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden...
and Frank H. Lutz - Lecture by David Gillman on The best picture of Poincare's homology sphere
- A cosmic hall of mirrors - physicsworld (26 Sep 2005)