Can the fundamental group of any manifold be realized as the fund grp of a finite space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:24:27Z http://mathoverflow.net/feeds/question/45549 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45549/can-the-fundamental-group-of-any-manifold-be-realized-as-the-fund-grp-of-a-finite Can the fundamental group of any manifold be realized as the fund grp of a finite space? Abhishek Parab 2010-11-10T12:52:35Z 2011-11-14T23:48:15Z <p>Recently, I was asked to calculate the fundamental group of the space $X= \{a,b,c,d\}$ with open sets generated by $\{ a, c, abc, acd \}$.</p> <p>Turns out, $\pi_1(X)\cong \mathbb Z$ and in fact, $X$ is the quotient of $S^1$ (with the northern and southern hemispheres identified). But the result was not so easy to prove and this motivates the questions:</p> <ul> <li><p>Is the fundamental group of every connected manifold (other restrictions / generalizations on the manifold are welcome) the fundamental group of a finite space? (Of course, it would not be Hausdorff). (I observe that there are many redundant points on a manifold where homotopy-equivalent loops need not traverse)</p></li> <li><p>Is calculating $\pi_1$ of such finite spaces easier than for the given space? (If yes, this gives a method to calculate fundamental groups of many familiar spaces)</p></li> </ul> <p>Perhaps the fact that -- $\pi_1$ of any CW complex just depends on its 1-skeleton [edit:2 skeleton]-- may be helpful.</p> http://mathoverflow.net/questions/45549/can-the-fundamental-group-of-any-manifold-be-realized-as-the-fund-grp-of-a-finite/45550#45550 Answer by Dan Petersen for Can the fundamental group of any manifold be realized as the fund grp of a finite space? Dan Petersen 2010-11-10T13:01:40Z 2010-11-10T13:01:40Z <p>In fact, there is the following <strong>theorem</strong>: <em>Every finite CW complex is weakly homotopy equivalent to a finite topological space, and vice versa.</em> </p> <p>For simplicial complexes, this theorem is realized by mapping a complex to its face poset, and using the correspondence between finite posets and finite topological spaces. In the other direction, one maps a poset to its order complex. </p> <p>In general it is not easy to compute homotopy groups of a finite topological space. I know that there are some techniques in Jonathan Barmak's Ph.D. thesis.</p>