Fundamental groups and homology groups of closed subsets of the plane - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:31:04Z http://mathoverflow.net/feeds/question/110805 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110805/fundamental-groups-and-homology-groups-of-closed-subsets-of-the-plane Fundamental groups and homology groups of closed subsets of the plane Tina 2012-10-27T03:35:19Z 2012-10-27T13:38:36Z <p>Let $X$ be a closed subset of $\mathbb{R}^2$. What restrictions are there on $\pi_1(X)$ and on the homology groups of $X$ (both singular and Cech)? This is elementary if $X$ has reasonable local properties, but the example of the Hawaiian Earring shows that things can be very complicated indeed.</p> http://mathoverflow.net/questions/110805/fundamental-groups-and-homology-groups-of-closed-subsets-of-the-plane/110807#110807 Answer by Ryan Budney for Fundamental groups and homology groups of closed subsets of the plane Ryan Budney 2012-10-27T04:08:09Z 2012-10-27T04:08:09Z <p>Eda, K. Fundamental group of subsets of the plane. Topology and its Applications Volume 84, Issues 1-3, 24 April 1998, Pages 283-306</p> <p>This is more or less a duplicate of the math.se thread: <a href="http://math.stackexchange.com/questions/36279/the-fundamental-group-of-every-subset-of-mathbbr2-is-torsion-free" rel="nofollow">http://math.stackexchange.com/questions/36279/the-fundamental-group-of-every-subset-of-mathbbr2-is-torsion-free</a></p> http://mathoverflow.net/questions/110805/fundamental-groups-and-homology-groups-of-closed-subsets-of-the-plane/110830#110830 Answer by Jeremy Brazas for Fundamental groups and homology groups of closed subsets of the plane Jeremy Brazas 2012-10-27T13:38:36Z 2012-10-27T13:38:36Z <p>The fundamental group of a closed planar set naturally injects into the first Cech homotopy group, which is an inverse limit of finitely generated free groups. In particular, the algebraic restrictions gained are: the fundamental group must be locally free, fully residually free, and residually finite.</p> <p><a href="http://arxiv.org/abs/math/0512343" rel="nofollow">Fischer, H., Zastrow, A., The fundamental groups of subsets of closed surfaces inject into their first shape groups, Algebraic and Geometric Topology 5 (2005) 1655-1676.</a></p> <p>The first homology pro-group then consists of finitely generated free groups and the first Cech homology group is the inverse limit of these.</p> <p>When $X\subset \mathbb{R}^2$ is compact and locally path connected, the canonical map $H_1(X)\to \check{H}_1(X)$ is surjective but the kernel can be difficult to understand even for the <a href="http://jlms.oxfordjournals.org/content/62/1/305.short" rel="nofollow">Hawaiian earring.</a></p> <p><a href="http://130.44.194.100/proc/2000-128-05/S0002-9939-99-05670-1/S0002-9939-99-05670-1.pdf" rel="nofollow">Eda, K., Kawamura, K. The surjectivity of the canonical homomorphism from singular homology to Cech homology Proc. Amer. Math. Soc. 128 No. 5 (1999) 1487-1495</a></p>