Barratt-Milnor Sphere

In the mathematical sub-area of algebraic topology , the Barratt-Milnor spheres are an example of a compact finite-dimensional space whose homology groups do not vanish in arbitrarily high degrees and even have uncountable dimensions. They are named after Michael Barratt and John Milnor .

definition

The -dimensional Barratt-Milnor sphere can be defined as ${\displaystyle k}$

${\displaystyle X=\bigcup _{n\in \mathbb {N} }\left\{(x_{0},x_{1},\ldots ,x_{k})\in \mathbb {R} ^{k+1}:{\Bigl (}x_{0}-{\frac {1}{n}}{\Bigr )}^{2}+x_{1}^{2}+\ldots +x_{k}^{2}={\frac {1}{n^{2}}}\right\}}$.

It is therefore a countable union of spheres which have a single point in common and whose topology comes from a metric in which the diameter of the spheres converges towards zero as the diameter increases . ${\displaystyle k}$ ${\displaystyle n}$

For you get the Hawaiian earring . The term Barratt-Milnor Sphere is only used for . ${\displaystyle k=1}$ ${\displaystyle k>1}$

Homology groups

For the singular homology groups are not zero and even uncountable . ${\displaystyle q\equiv 1\ mod\ (r-1),q>1}$ ${\displaystyle H_{q}(X;\mathbb {Q} )}$

properties

The -dimensional Barratt-Milnor spheres are -contiguous and locally -contiguous . ${\displaystyle k}$ ${\displaystyle (k-1)}$ ${\displaystyle (k-1)}$

But they are not semilocal- related ${\displaystyle k}$ .

They are compact and dimensional. ${\displaystyle k}$

They are the one-point compactification of a countable union of s. ${\displaystyle \mathbb {R} ^{k}}$

literature

• M. Barratt, J. Milnor: An example of anomalous singular homology , Proc. Amer. Math. Soc. 13, 293-297 (1962)