Editing Homology sphere (section)

==Constructions and examples==

*[[Surgery theory|Surgery]] on a knot in the 3-sphere ''S''<sup>3</sup> with framing +1 or &minus;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 &minus;1.
*If ''p'', ''q'', and ''r'' are pairwise relatively prime positive integers then the link of the singularity ''x''<sup>''p''</sup> + ''y''<sup>''q''</sup> + ''z''<sup>''r''</sup> = 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 [[Egbert Brieskorn|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)]].)
*Suppose that <math>a_1, \ldots, a_r</math> are integers all at least 2 such that any two are coprime. Then the [[Seifert fiber space]]

:: <math>\{b, (o_1,0);(a_1,b_1),\dots,(a_r,b_r)\}\,</math>

:over the sphere with exceptional fibers of degrees ''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub> is a homology sphere, where the ''b'''s are chosen so that

:: <math>b+b_1/a_1+\cdots+b_r/a_r=1/(a_1\cdots a_r).</math>

:(There is always a way to choose the ''b''&prime;s, and the homology sphere does not depend (up to isomorphism) on the choice of ''b''&prime;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''&prime;s are 2, 3, and 5 this gives the Poincaré sphere. If there are at least 3 ''a''&prime;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)|SL<sub>2</sub>('''R''')]].