The properties of space that survive stretching but not tearing.
11 topics · 18 curated works
No prior grounding assumed.
Topology Without Tears
Sidney A. Morris · 1989
Builds point-set topology from metric spaces toward the general definition of a topological space, arguing each axiom should be motivated by a…
+1 more at this level
Assumes you know the vocabulary.
Topology and Data
Gunnar Carlsson · 2009
Argues that the shape of high-dimensional data — its clusters, loops and voids — is a legitimate object of study in its own right, and adapts…
+2 more at this level
Primary sources and full treatments.
Analysis Situs
Henri Poincaré · 1895
Founds algebraic topology by defining the fundamental group and a numerical invariant, the Betti numbers, that survive continuous deformation,…
+12 more at this level
12 of 18 works
Sidney A. Morris
Builds point-set topology from metric spaces toward the general definition of a topological space, arguing each axiom should be motivated by a concrete example before it is stated abstractly.
link checked 17 Sept 2026Colin C. Adams
Introduces knot theory through diagrams and hands-on invariants rather than algebraic machinery, arguing the subject is learnable, and full of open problems, without a graduate topology course first.
Gunnar Carlsson
Argues that the shape of high-dimensional data — its clusters, loops and voids — is a legitimate object of study in its own right, and adapts homology to work on point clouds instead of manifolds.
link checked 17 Sept 2026Robert Ghrist
Argues that topology, not just calculus, is the natural mathematics of large-scale qualitative structure, building the case through persistent homology, sheaves and Euler calculus applied to sensor networks, signals and data.
link checked 17 Sept 2026Daniel Rosiak
Teaches sheaf theory, the machinery for gluing locally defined data into a consistent global structure, by building outward from dozens of worked examples across mathematics and science rather than from category-theoretic axioms first.
link checked 17 Sept 2026Henri Poincaré
Founds algebraic topology by defining the fundamental group and a numerical invariant, the Betti numbers, that survive continuous deformation, arguing a space's connectivity can be studied algebraically rather than only geometrically.
link checked 17 Sept 2026James W. Alexander
Constructs a polynomial invariant from a knot's diagram that is provably unchanged by Reidemeister moves, giving the first computable tool for telling two knots apart.
link checked 17 Sept 2026J. H. C. Whitehead
Introduces CW complexes as the right combinatorial setting for homotopy theory, arguing that cellular attachment, not simplicial rigidity, is what makes a space's homotopy type computable.
link checked 17 Sept 2026John Milnor
Shows that the topology of a manifold is controlled by the critical points of any sufficiently generic smooth function on it, turning the classification of manifolds into a study of critical-point behaviour.
William P. Thurston
Argues that most three-manifolds carry one of eight geometric structures, and develops hyperbolic geometry as the generic and richest of them rather than as a special case.
link checked 17 Sept 2026Raoul Bott
Surveys how the idea that a function's critical points control a manifold's topology grew from Morse's original inequalities into later applications in gauge theory and loop spaces.
link checked 17 Sept 2026Vaughan F. R. Jones
Discovers a new knot polynomial by an unexpected route through operator algebras, arguing that representations of the braid group inside a von Neumann algebra carry more information about a link's isotopy type than any diagram-based invariant then known.
link checked 17 Sept 2026