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AbeBooks Bookseller Since: May 31, Home R. Sher, R. Daverman Handbook of Geometric Topology. Stock Image. Handbook of Geometric Topology R. Published by North Holland, New Condition: New Hardcover. Save for Later. Discrete Fractional Calculus.
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Handbook of Geometric Topology
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Emily Riehl. The terminology "geometric topology" as far as I'm aware is a fairly recent historical phenomenon. The words used by topologists to describe their areas has had a fair bit of flux over the years. Before the mid's, algebraic topology was called combinatorial topology. I think the urge to use the phrase geometric topology began sometime after the advent of the h-cobordism theorem, and the observation that high-dimensional manifold theory, via a rather elaborate formulation can be largely turned into elaborate algebraic problems.
So there was a desire to have a term that held-together all the aspects of topology where these techniques either don't apply, or were not used or at least, not predominantly used. That's geometric topology. So 2, 3 and 4-dimensional manifold theory would be a big chunk of this area. But of course, even if high-dimensional manifold theory in principle reduces to algebra, that doesn't necessarily mean you want to use that reduction -- it may be too complicated to be useful.
So there are higher-order type high-dimensional manifold theory problems that don't fit the traditional reductions. Like say Vassiliev's work on spaces of knots. So this would also be considered geometric topology. Defining a subject by what it's not is kind of strange and artificial but all these subject-area definitions are kind of strange and artificial. I think the above-quoted blurbs also get at a key aspect of the area. Algebraic topology tends to be more focused on a broad set of tools.