In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a homology sphere is an ''n''-
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
''X'' having the
homology group
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...
s of an ''n''-
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, for some integer
. That is,
:
and
:
for all other ''i''.
Therefore ''X'' is a
connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
, with one non-zero higher
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 simplicia ...
, namely,
. It does not follow that ''X'' is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
, only that its
fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
is
perfect (see
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. The theorem is named after Witold Hurewicz, and generalizes earlier results ...
).
A
rational homology sphere is defined similarly but using homology with rational coefficients.
Poincaré homology sphere
The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere, first constructed by
Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
. Being a
spherical 3-manifold, it is the only homology 3-sphere (besides the
3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior o ...
itself) with a finite
fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
. Its fundamental group is known as the
binary icosahedral group
In mathematics, the binary icosahedral group 2''I'' or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 120.
It is an extension of the icosahedral group ''I'' or (2,3,5) o ...
and has order 120. Since the fundamental group of the 3-sphere is trivial, this shows that there exist 3-manifolds with the same homology groups as the 3-sphere that are not homeomorphic to it.
Construction
A simple construction of this space begins with a
dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
. Each face of the dodecahedron is identified with its opposite face, using the minimal clockwise twist to line up the faces.
Gluing each pair of opposite faces together using this identification yields a closed 3-manifold. (See
Seifert–Weber space In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discov ...
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)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a ...
/I where I is the
icosahedral group
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of th ...
(i.e., the rotational
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
of the regular
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
and dodecahedron, isomorphic to the
alternating group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...
A
5). 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
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ...
of SO(3) which can be realized as the group of unit
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s and is
homeomorphic
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the 3-sphere. In this case, the Poincaré homology sphere is isomorphic to
where
is the
binary icosahedral group
In mathematics, the binary icosahedral group 2''I'' or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 120.
It is an extension of the icosahedral group ''I'' or (2,3,5) o ...
, the perfect
double cover of I
embedded in
.
Another approach is by
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: ''drilling'' then '' ...
. The Poincaré homology sphere results from +1 surgery on the right-handed
trefoil knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot (mathematics), knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop (topology ...
.
Cosmology
In 2003, lack of structure on the largest scales (above 60 degrees) in the
cosmic microwave background
The cosmic microwave background (CMB, CMBR), or relic radiation, is microwave radiation that fills all space in the observable universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dar ...
as observed for one year by the
WMAP
The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP and Explorer 80), was a NASA spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic mic ...
spacecraft led to the suggestion, by
Jean-Pierre Luminet
Jean-Pierre Luminet (born 3 June 1951) is a French astrophysicist, specializing in black holes and cosmology. He is an emeritus research director at the CNRS ( Centre national de la recherche scientifique). Luminet is a member of the Laboratoir ...
of the
Observatoire de Paris and colleagues, that the
shape of the universe
In physical cosmology, the shape of the universe refers to both its local and global geometry. Local geometry is defined primarily by its curvature, while the global geometry is characterised by its topology (which itself is constrained by curv ...
is a Poincaré sphere.
["Is the universe a dodecahedron?"](_blank)
article at PhysicsWorld. 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.
Data analysis from the
Planck spacecraft suggests that there is no observable non-trivial topology to the universe.
Constructions and examples
*
Surgery
Surgery is a medical specialty that uses manual and instrumental techniques to diagnose or treat pathological conditions (e.g., trauma, disease, injury, malignancy), to alter bodily functions (e.g., malabsorption created by bariatric surgery s ...
on a knot in the 3-sphere ''S''
3 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 ''x''
''p'' + ''y''
''q'' + ''z''
''r'' = 0 (in other words, the intersection of a small 3-sphere around 0 with this complex surface) is a
Brieskorn manifold that 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
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 ...
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)
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 ...
.)
*Suppose that
are integers all at least 2 such that any two are coprime. Then the
Seifert fiber space
::
:over the sphere with exceptional fibers of degrees ''a''
1, ..., ''a''
''r'' is a homology sphere, where the ''bs are chosen so that
::
:(There is always a way to choose the ''b''′s, and the homology sphere does not depend (up to isomorphism) 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
-valued invariant of homology 3-spheres.
*The
Casson invariant 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 manifold that is not a
topological manifold
In topology, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds ...
. The double suspension of ''A'' is homeomorphic to the standard 5-sphere, but its
triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle m ...
(induced by some triangulation of ''A'') is not a
PL manifold
In mathematics, a piecewise linear manifold (PL manifold) is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas (topology), atlas, such that one can pass from chart (topolog ...
. In other words, this gives an example of a finite
simplicial complex
In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
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
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
there is a homology 3 sphere Σ with
Rokhlin invariant 1 such that the
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 ...
Σ#Σ of Σ with itself bounds a smooth acyclic 4-manifold.
Ciprian Manolescu
Ciprian Manolescu (; born December 24, 1978) is a Romanian-American mathematician, working in gauge theory, symplectic geometry, and low-dimensional topology. He is currently a professor of mathematics at Stanford University.
Biography
Manolescu ...
showed that there is no such homology sphere with the given property, and therefore, there are 5-manifolds not homeomorphic to simplicial complexes. In particular, the example originally given by Galewski and Stern
is not triangulable.
See also
*
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders 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. ...
*
Moore space (algebraic topology)
References
Selected reading
*
*
*
Robion Kirby
Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he developed the Kirby–Siebenmann invariant ...
, Martin Scharlemann, ''Eight faces of the Poincaré homology 3-sphere''. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 113–146,
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal complete ...
, New York-London, 1979.
*
* 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 Verticesby
Anders Björner
Anders Björner (born 17 December 1947) is a Swedish professor of mathematics, in the Department of Mathematics at the Royal Institute of Technology, Stockholm, Sweden. He received his Ph.D. from Stockholm University in 1979, under Bernt Lindströ ...
and
Frank H. Lutz
*Lecture by
David Gillman o
The best picture of Poincare's homology sphere
Topological spaces
Homology theory
3-manifolds
Spheres