The ring of integers looks like the 3-dimensional sphere viewed as the Hopf fibration - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:48:20Z http://mathoverflow.net/feeds/question/30186 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/30186/the-ring-of-integers-looks-like-the-3-dimensional-sphere-viewed-as-the-hopf-fib The ring of integers looks like the 3-dimensional sphere viewed as the Hopf fibration Ariyan Javanpeykar 2010-07-01T13:48:46Z 2010-07-01T14:07:58Z <p>This question is based on the following phrase:</p> <p>"In a sense, $\textrm{Spec} \ \mathbf{Z}$ looks topologically like a 3-dimensional sphere viewed as the Hopf fibration over $\mathbf{S}^2$."</p> <p>See page 88 of <em>Algebraic Geometry II</em> by Shafarevich.</p> <p>I find this remark very interesting but I can't seem to parse it. </p> <p>I always just viewed $\textrm{Spec} \ \mathbf{Z}$ as an arithmetic analogue of $\mathbf{P}^1(\mathbf{C}) = \mathbf{S}^2$. This remark would add "something" to that in a sense.</p> http://mathoverflow.net/questions/30186/the-ring-of-integers-looks-like-the-3-dimensional-sphere-viewed-as-the-hopf-fib/30187#30187 Answer by David Corfield for The ring of integers looks like the 3-dimensional sphere viewed as the Hopf fibration David Corfield 2010-07-01T14:07:58Z 2010-07-01T14:07:58Z <p>Various pieces of exposition and references are to be found - <a href="http://golem.ph.utexas.edu/category/2007/10/this_weeks_finds_in_mathematic_18.html" rel="nofollow">here</a>, <a href="http://math.ucr.edu/home/baez/week257.html" rel="nofollow">here</a>, <a href="http://www.ucl.ac.uk/~ucahmki/baez13.12.pdf" rel="nofollow">here</a>, and <a href="http://golem.ph.utexas.edu/category/2009/04/afternoon_fishing.html" rel="nofollow">here</a>.</p>