Homology sphere


In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer. That is,
and
Therefore X is a connected space, with one non-zero higher Betti number, namely,. It does not follow that X is simply connected, only that its fundamental group is perfect.
A rational homology sphere is defined similarly but using homology with rational coefficients.

Poincaré homology sphere

The Poincaré homology sphere is a particular example of a homology sphere, first constructed by Henri Poincaré. Being a spherical 3-manifold, it is the only homology 3-sphere with a finite fundamental group. Its fundamental group is known as the binary icosahedral group 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. 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.
Alternatively, the Poincaré homology sphere can be constructed as the quotient space SO/I where I is the icosahedral group. More intuitively, this means that the Poincaré homology sphere is the space of all geometrically distinguishable positions of an icosahedron in Euclidean 3-space. One can also pass instead to the universal cover of SO which can be realized as the group of unit quaternions and is homeomorphic to the 3-sphere. In this case, the Poincaré homology sphere is isomorphic to where is the binary icosahedral group, the perfect double cover of I embedded in.
Another approach is by Dehn surgery. The Poincaré homology sphere results from +1 surgery on the right-handed trefoil knot.

Cosmology

In 2003, lack of structure on the largest scales in the cosmic microwave background as observed for one year by the WMAP 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.
Data analysis from the Planck spacecraft suggests that there is no observable non-trivial topology to the universe.

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 and framings 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 is a Brieskorn manifold that is a homology 3-sphere, called a Brieskorn 3-sphere Σ. It is homeomorphic to the standard 3-sphere if one of p, q, and r is 1, and Σ 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.
  • Suppose that are integers all at least 2 such that any two are coprime. Then the Seifert fiber space

    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. The double suspension of A is homeomorphic to the standard 5-sphere, but its triangulation 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.
Galewski and Stern showed that all compact topological manifolds 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. Ciprian 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.

Selected reading