Sphere of homology

A homology sphere referred to in the mathematics one -dimensional manifold whose singular homology groups to those of ordinary isomorphic are -sphere. ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle n}$

definition

Expressed explicitly, a -dimensional manifold is called for its singular homology groups ${\displaystyle n}$ ${\displaystyle M}$

${\displaystyle H_{0}(M,\mathbb {Z} )\cong H_{n}(M,\mathbb {Z} )\cong \mathbb {Z} }$

for one and ${\displaystyle n>1}$

${\displaystyle H_{j}(M,\mathbb {Z} )=\{0\}}$

apply to all others . ${\displaystyle j}$

From the homology one can see that a compact, connected manifold has no boundary. In general, however, it is not simply connected : If you divide the fundamental group by its commutator group, you get a group that is isomorphic to the first homology group . This means from one can only conclude that the fundamental group is a perfect group , i.e. isomorphic to its commutator group , but not that it has to be trivial. ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \pi _{1}(M)}$ ${\displaystyle H_{1}(M,\mathbb {Z} )}$ ${\displaystyle H_{1}(M,\mathbb {Z} )=\{0\}}$ ${\displaystyle \pi _{1}(M)}$

Historical classification

Historically, spheres of homology were first considered in the -dimensional topology. ${\displaystyle 3}$

Poincaré initially believed that the homology ring must be sufficient to uniquely characterize the standard -dimensional sphere. However, he discovered a counterexample (the so-called Poincaré homology sphere ) and then formulated the more stringent Poincaré conjecture (which is also required), which Perelman only proved about 100 years later . ${\displaystyle 3}$ ${\displaystyle \pi _{1}(M)=\{0\}}$

Connection to the homotopy sphere

Application of Hurewicz's theorem and Whitehead's theorem shows that every simply connected -dimensional homology sphere is a homotopy sphere , i. H. must be homotopy equivalent to the sphere . From the Poincaré conjecture or its higher-dimensional analogue for it follows that it is also homeomorphic to . So even in higher dimensions there are spheres of homology only for . ${\displaystyle n}$ ${\displaystyle S^{n}}$ ${\displaystyle n>3}$ ${\displaystyle S^{n}}$ ${\displaystyle M\not =S^{n}}$ ${\displaystyle \pi _{1}M\not =0}$

Web links

• Classification of homology 3-spheres? (mathoverflow)