The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key example is the literature surrounding the Poincaré Conjecture.

The terminology "geometric topology" is fairly recent.

The words used by topologists to describe their areas has had a fair bit of flux over the years. Before the mid-40's, algebraic topology was called combinatorial topology. 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). Thus a big chunk of "geometric topology" is concerned with 2, 3 and 4-dimensional manifold theory. But of course, even if high-dimensional manifold theory in principle reduces to algebra, that doesn't necessarily mean that the reduction is the right tool to use -- it may be too complicated to be useful. These higher-order type high-dimensional manifold theory problems that don't fit the traditional reductions -- like say Vassiliev's work on spaces of knots -- also end up under the banner of geometric topology.

Defining a subject by what it's not is kind of strange and artificial, but so is taxonomy in general. To again compare it with algebraic topology, note that algebraic topology tends to be more focused on a broad set of tools. Geometric topology, on the other hand, is focused more by the goals, things like the Poincare conjecture(s) and such. So the latter tends to have a more example-oriented culture.

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