Geometric topology
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In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. It has come over time to be almost synonymous with low-dimensional topology, concerning in particular manifolds of four or fewer dimensions.
In the rapid development of topology after 1945, a distinction was drawn between the fields of algebraic topology typified by homotopy theory, geometric topology with the Poincaré conjecture as its biggest unsolved problem, and differential topology as the study mostly of differential structures, with Morse theory and transversality as their natural techniques. These fields all rested on general topology, which was the study of the general topological space. This classification would come to seem more artificial, with the passing of years.
A number of advances starting in the 1960s had the effect of changing geometric topology. The solution by Smale, in 1961, of the Poincaré conjecture in higher dimensions made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory. Thurston's geometrization conjecture, formulated in the late 1970s, offered a framework that suggested geometry and topology were closely intertwined in low dimensions, and Thurston's proof of geometrization for Haken manifolds utilized a variety of tools from previously only weakly linked areas of mathematics. Vaughan Jones' discovery of the Jones polynomial in the early 1980s not only led knot theory in new directions but gave rise to still mysterious connections between low-dimensional topology and mathematical physics. In 2002 Grigori Perelman announced a proof of the three-dimensional Poincare conjecture, using Richard Hamilton's Ricci flow: an idea belonging to the field of geometric analysis.
Overall, this progress has led to better integration of the field into the rest of mathematics.
[edit] See also
[edit] External links
- Rob Kirby's Problems in Low-Dimensional Topology -gzipped postscript file (1.4MB)
- Mark Brittenham's links to low dimensional topology - lists of homepages, conferences, etc.
