Homology

Reference

Sources & Further Reading

The books, papers, lectures and explainers this book stands on — and where to go next.

If you read only three more things

Gentle, generous, and free — the best next steps after this book.

  • Algebraic Topology — Allen Hatcher (2002) free
    The standard modern textbook, free online. Chapter 2 (homology) and Chapter 3 (cohomology) open with exactly the graph-and-loop intuitions used here.
  • Hodge Laplacians on Graphs — Lek-Heng Lim (SIAM Review, 2020) free
    Cohomology and Hodge theory as pure linear algebra on graphs — readable with nothing more than matrices.
  • Barcodes: the persistent topology of data — Robert Ghrist (Bull. AMS, 2008) free
    Fifteen pages that explain why homology became a tool for data science. Beautifully written.

Textbooks on homology and cohomology

From the friendliest to the most complete.

  • Graphs, Surfaces and Homology — Peter Giblin (CUP, 3rd ed. 2010)
    The gentlest real textbook: graphs, then surfaces, then homology, with a mod-2 chapter and hundreds of exercises.
  • Elementary Applied Topology — Robert Ghrist (2014)
    A visual, application-first tour: homology, cohomology, sheaves and persistence, with hundreds of figures.
  • Elements of Algebraic Topology — James Munkres (1984)
    The careful classic for simplicial homology and its conventions.
  • Differential Forms in Algebraic Topology — Raoul Bott & Loring Tu (Springer, 1982)
    The gold standard for de Rham cohomology, Mayer–Vietoris, Čech cohomology and characteristic classes.
  • Characteristic Classes — John Milnor & James Stasheff (Princeton, 1974)
    The reference for Stiefel–Whitney, Euler and Chern classes.
  • Homology, Cohomology, and Sheaf Cohomology — Jean Gallier & Jocelyn Quaintance
    A long, careful set of notes that goes all the way to sheaf cohomology.
  • Mod Two Homology and Cohomology — Jean-Claude Hausmann (Springer Universitext)
    Develops the whole theory with ℤ/2 coefficients — and argues convincingly that this is the best way to start.

Foundations: proofs, algebra and linear algebra

For the prerequisites taught in Parts 0 and I.

  • Book of Proof — Richard Hammack free
    Logic, sets, functions, relations and proof techniques, from zero. Free.
  • How to Read Mathematics — Shai Simonson & Fernando Gouvêa free
    A short guide to the slow, active way mathematics has to be read.
  • Abstract Algebra: Theory and Applications — Thomas Judson free
    A complete, free algebra textbook: groups, cosets, quotients, homomorphisms.
  • Visual Group Theory and Group Explorer — Nathan Carter free
    Groups through Cayley diagrams; the free web app lets you play with hundreds of groups.
  • Expository papers (“blurbs”) — Keith Conrad free
    Short, crystal-clear notes; “Quotient Groups” is the one to read with Chapter 1.4.
  • Math3ma — Tai-Danae Bradley free
    Warm, intuitive essays on quotient groups, the isomorphism theorems and category theory.
  • Essence of Linear Algebra — 3Blue1Brown (Grant Sanderson) free
    Matrices as transformations of space, kernels, ranks and duality — animated.
  • Immersive Linear Algebra — J. Ström, K. Åström & T. Akenine-Möller free
    An interactive linear algebra book with live figures in every chapter.
  • Interactive Linear Algebra — Dan Margalit & Joseph Rabinoff free
    A free textbook with interactive demonstrations of row reduction, span and rank.

Topology, homotopy and manifolds

For Part II.

  • Topology Without Tears — Sidney Morris free
    A free, patient introduction to point-set topology.
  • The Shape of Space, and Torus Games — Jeffrey Weeks free
    Flat tori, Klein bottles and gluing diagrams made playable — the inspiration for several figures here.
  • Conway’s ZIP proof — George Francis & Jeffrey Weeks (Amer. Math. Monthly, 1999)
    A short, beautiful proof of the classification of surfaces.
  • Euler’s Gem — David Richeson (Princeton, 2008)
    The popular history of V − E + F, from Euler to modern topology.
  • Proofs and Refutations — Imre Lakatos (1976)
    A dialogue about Euler’s formula, its counterexamples, and how mathematics is really made.

Differential forms and de Rham cohomology

For Chapters 4.3–4.4.

  • A Geometric Approach to Differential Forms — David Bachman free
    Forms explained through pictures and geometry first, formulas second.
  • Discrete Differential Geometry: An Applied Introduction — Keenan Crane free
    Discrete forms as integrated values on meshes; the most practical bridge between cochains and forms.
  • Differential forms and integration — Terence Tao free
    A short essay on what forms really are and why they are integrated.
  • A Visual Introduction to Differential Forms and Calculus on Manifolds — Jon Pierre Fortney (Birkhäuser, 2018)
    Hundreds of pictures; written for readers meeting forms for the first time.
  • Visual Differential Geometry and Forms — Tristan Needham (Princeton, 2021)
    A lavishly illustrated, geometric account of curvature and forms.
  • Discrete Exterior Calculus — M. Desbrun, A. Hirani, M. Leok & J. Marsden free
    Cochains as discrete differential forms, with a discrete Hodge star.

Cohomology in the wild

Impossible figures, rankings, sheaves — cohomology as the measure of local-to-global failure.

Knots and links

For Chapter 3.7: from colourings and braids to Khovanov homology.

  • The Knot Book — Colin C. Adams (1994; AMS reprint 2004)
    The friendliest introduction there is: diagrams, tabulation, invariants, braids and the Jones polynomial, with open problems and pictures on almost every page.
  • Knots and Links — Dale Rolfsen (1976)
    The classic geometric treatment, with Seifert surfaces, covering spaces and the Alexander polynomial done properly, and the knot table that everyone still uses.
  • An Introduction to Knot Theory — W. B. Raymond Lickorish (1997)
    A concise graduate text: Seifert matrices, the Jones polynomial and Tait’s conjectures, and the 3-manifold side of the story.
  • On Khovanov’s categorification of the Jones polynomial — Dror Bar-Natan (Algebraic & Geometric Topology, 2002) free
    The paper that made Khovanov homology computable and popular: the cube of resolutions explained clearly, with tables for all prime knots up to eleven crossings.
  • KnotInfo and the Knot Atlas — Charles Livingston and Allison H. Moore; Dror Bar-Natan, Scott Morrison and others free
    Two searchable tables of knots with their diagrams, braids, polynomials, genus and unknotting numbers; the Knot Atlas (katlas.org) adds Khovanov homology.

Persistent homology and data

For Chapter 3.8 and beyond.

  • Topology and data — Gunnar Carlsson (Bull. AMS, 2009)
    The manifesto of topological data analysis.
  • What is… persistent homology? — Shmuel Weinberger (Notices AMS, 2011)
    Two pages of perfect intuition.
  • A roadmap for the computation of persistent homology — N. Otter, M. Porter, U. Tillmann, P. Grindrod & H. Harrington (2017) free
    The full pipeline, software, and a hands-on tutorial.
  • An introduction to Topological Data Analysis — Frédéric Chazal & Bertrand Michel (2021) free
    Rips and Čech complexes, stability, and practice, for data scientists.
  • Computational Topology: An Introduction — Herbert Edelsbrunner & John Harer (AMS, 2010)
    Algorithms, the elder rule, persistence diagrams and stability.
  • Persistent Cohomology and Circular Coordinates — V. de Silva, D. Morozov & M. Vejdemo-Johansson (2011) free
    Turning cohomology classes of data into angle-valued coordinates.
  • Computing homology (and other posts) — Jeremy Kun, Math ∩ Programming free
    Homology computed in Python with row reduction — a programmer’s introduction.

Categories and homological algebra

For Part V.

Lectures to watch

Recommended lecture series and explainers (listed from their course pages and descriptions).

History

How the ideas came to be.

  • Papers on Topology: Analysis Situs and Its Five Supplements — Henri Poincaré, translated by John Stillwell (AMS/LMS, 2010)
    Where homology began, in 1895 — and where the word “torsion” comes from.
  • Emmy Noether and Topology — Friedrich Hirzebruch (1996 lecture) free
    How Noether turned Betti numbers into groups, mostly in conversation.
  • In Search of Shadows: the First Topological Conference, Moscow 1935 — D. Apushkinskaya, A. Nazarov & G. Sinkevich free
    The conference where cohomology and the cup product appeared.

On explaining mathematics

The craft this book tries to practise.

Works cited

Every work the chapters cite, with the chapters that cite it.

  1. Michał Adamaszek and Henry Adams (2017). The Vietoris–Rips complexes of a circle. Pacific Journal of Mathematics 290(1), 1–40. free online 3.8
  2. J. Frank Adams and Michael F. Atiyah (1966). K-theory and the Hopf invariant. Quarterly Journal of Mathematics 17(1), 31–38. doi 5.3
  3. Yakir Aharonov and David Bohm (1959). Significance of electromagnetic potentials in the quantum theory. Physical Review 115, 485–491. doi 4.4
  4. Selman Akbulut and Robion Kirby (1985). A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews–Curtis conjecture. Topology 24 (1985), 375–390. doi 2.3
  5. Aleph 0 (2025). What is algebraic topology?. YouTube video, 10 February 2025 (14 minutes). free online 3.3
  6. J. W. Alexander (1922). A proof and extension of the Jordan–Brouwer separation theorem. Transactions of the American Mathematical Society 23(4), 333–349. free online 3.2
  7. James W. Alexander (1923). A lemma on systems of knotted curves. Proceedings of the National Academy of Sciences 9 (1923), 93–95. free online 3.7
  8. J. W. Alexander (1924). An example of a simply connected surface bounding a region which is not simply connected. Proceedings of the National Academy of Sciences 10(1), 8–10. free online 3.5
  9. James W. Alexander (1928). Topological invariants of knots and links. Transactions of the American Mathematical Society 30(2), 275–306. doi 3.74.6
  10. James W. Alexander and Garland B. Briggs (1927). On types of knotted curves. Annals of Mathematics 28 (1926–27), 562–586. doi 3.7
  11. Paul Alexandroff (1928). Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung. Mathematische Annalen 98, 617–635. doi 4.7
  12. Josh Alman, Ran Duan, Virginia Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Renfei Zhou (2025). More asymmetry yields faster matrix multiplication. Proceedings of the 2025 ACM–SIAM Symposium on Discrete Algorithms (SODA). free online 3.4
  13. Levent Alpöge (2026). The (3, 4, ∞) modular family of 2-tori, completed at its three special points, is a complex structure on S⁶. manuscript, August 2026, written with the AI model Claude. free online 4.8
  14. Marlow Anderson and Todd Feil (1998). Turning Lights Out with linear algebra. Mathematics Magazine 71(4), 300–303. doi 1.5
  15. James J. Andrews and Morton L. Curtis (1965). Free groups and handlebodies. Proceedings of the American Mathematical Society 16 (1965), 192–195. free online 2.3
  16. Darya E. Apushkinskaya, Alexander I. Nazarov and Galina I. Sinkevich (2019). In search of shadows: the First Topological Conference, Moscow 1935. The Mathematical Intelligencer 41, 37–42. free online 4.24.5
  17. Aristotle (c. 350 BC). Prior Analytics. translated by A. J. Jenkinson. free online 0.2
  18. M. A. Armstrong (1983). Basic Topology. Springer, Undergraduate Texts in Mathematics. doi 2.6
  19. Emil Artin (1925). Theorie der Zöpfe. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 4 (1925), 47–72. doi 3.7
  20. Emil Artin (1947). Theory of braids. Annals of Mathematics 48 (1947), 101–126. doi 3.7
  21. Michael F. Atiyah and Friedrich Hirzebruch (1961). Vector bundles and homogeneous spaces. Proceedings of Symposia in Pure Mathematics 3, 7–38. doi 5.3
  22. Michael F. Atiyah (1988). Topological quantum field theory. Publications Mathématiques de l’IHÉS 68, 175–186. free online 5.3
  23. Michael Atiyah (2001). Mathematics in the 20th century. American Mathematical Monthly 108(7), 654–666. doi 3.4
  24. Sheldon Axler (2024). Linear Algebra Done Right (4th edition). Springer, Undergraduate Texts in Mathematics (open access). free online 1.5
  25. László Babai (2016). Graph isomorphism in quasipolynomial time. Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC 2016), 684–697. free online 1.2
  26. John Baez (2010). This Week’s Finds in Mathematical Physics (Week 293). online column, 5 February 2010. free online 4.1
  27. Dror Bar-Natan (2002). On Khovanov’s categorification of the Jones polynomial. Algebraic & Geometric Topology 2 (2002), 337–370. free online 3.7
  28. Ulrich Bauer (2021). Ripser: efficient computation of Vietoris–Rips persistence barcodes. Journal of Applied and Computational Topology 5, 391–423. free online 3.8
  29. Michael V. Berry (1984). Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London A 392, 45–57. doi 4.8
  30. David Farrell (2021). You Could Have Invented Homology, Part 3: Boundaries & The Big Idea. Boarbarktree, YouTube video (16 February 2021); part 1 appeared on 16 December 2020. free online 2.13.2
  31. William W. Boone (1958). The word problem. Proceedings of the National Academy of Sciences 44(10), 1061–1065. free online 2.3
  32. Armand Borel and Jean-Pierre Serre (1953). Groupes de Lie et puissances réduites de Steenrod. American Journal of Mathematics 75 (1953), 409–448. doi 4.8
  33. Karol Borsuk (1948). On the imbedding of systems of compacta in simplicial complexes. Fundamenta Mathematicae 35, 217–234. doi 3.84.7
  34. Raoul Bott (1959). The stable homotopy of the classical groups. Annals of Mathematics 70(2), 313–337. doi 5.3
  35. Raoul Bott and Loring W. Tu (1982). Differential Forms in Algebraic Topology. Springer, Graduate Texts in Mathematics 82. doi 4.44.7
  36. Werner Boy (1903). Über die Curvatura integra und die Topologie geschlossener Flächen. Mathematische Annalen 57, 151–184. doi 2.2
  37. Tai-Danae Bradley (2016). What’s a quotient group, really? Part 2. Math3ma (blog), 22 November 2016. free online 1.4
  38. Simon Brendle and Pei-Ken Hung (2026). A metric on S² × S² with positive sectional curvature. preprint. free online 4.8
  39. Mark Brittenham and Susan Hermiller (2026). Unknotting number is not additive under connected sum. Annals of Mathematics 204 (2026), 795–817. doi 3.7
  40. L. E. J. Brouwer (1911). Über Abbildung von Mannigfaltigkeiten. Mathematische Annalen 71, 97–115. doi 0.13.55.1
  41. L. E. J. Brouwer (1911). Beweis der Invarianz der Dimensionenzahl. Mathematische Annalen 70, 161–165. doi 2.43.5
  42. Edgar H. Brown (1962). Cohomology theories. Annals of Mathematics 75(3), 467–484. doi 5.3
  43. William Burnside (1902). On an unsettled question in the theory of discontinuous groups. Quarterly Journal of Pure and Applied Mathematics 33 (1902), 230–238. 1.3
  44. Benjamin A. Burton (2020). The next 350 million knots. 36th International Symposium on Computational Geometry (SoCG 2020), LIPIcs 164, 25:1–25:17. free online 3.7
  45. Lewis Campbell and William Garnett (1882). The Life of James Clerk Maxwell. Macmillan, London. free online 3.7
  46. Georg Cantor (1878). Ein Beitrag zur Mannigfaltigkeitslehre. Journal für die reine und angewandte Mathematik 84, 242–258. doi 1.13.5
  47. Gunnar Carlsson, Tigran Ishkhanov, Vin de Silva and Afra Zomorodian (2008). On the local behavior of spaces of natural images. International Journal of Computer Vision 76(1), 1–12. doi 3.85.3
  48. Élie Cartan (1899). Sur certaines expressions différentielles et le problème de Pfaff. Annales scientifiques de l’École Normale Supérieure, 3e série, 16, 239–332. free online 4.3
  49. Augustin-Louis Cauchy (1813). Recherches sur les polyèdres. Premier mémoire. Journal de l’École polytechnique 9 (cahier 16), 68–86; read to the Institut in February 1811. 2.6
  50. Arthur Cayley (1854). On the theory of groups, as depending on the symbolic equation θⁿ = 1. The London, Edinburgh, and Dublin Philosophical Magazine 7(42), 40–47. doi 1.3
  51. Eduard Čech (1932). Théorie générale de l’homologie dans un espace quelconque. Fundamenta Mathematicae 19, 149–183. doi 4.7
  52. Joseph Minhow Chan, Gunnar Carlsson and Raul Rabadan (2013). Topology of viral evolution. Proceedings of the National Academy of Sciences 110(46), 18566–18571. free online 3.85.3
  53. Frédéric Chazal, Vin de Silva and Steve Oudot (2014). Persistence stability for geometric complexes. Geometriae Dedicata 173, 193–214. free online 3.8
  54. Frédéric Chazal and Bertrand Michel (2021). An introduction to topological data analysis: fundamental and practical aspects for data scientists. Frontiers in Artificial Intelligence 4, 667963. free online 3.8
  55. Shiing-Shen Chern (1944). A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds. Annals of Mathematics 45(4), 747–752. doi 4.8
  56. Shiing-Shen Chern (1946). Characteristic classes of Hermitian manifolds. Annals of Mathematics 47(1), 85–121. doi 4.8
  57. Timothy Y. Chow (2006). You could have invented spectral sequences. Notices of the American Mathematical Society 53(1), 15–19. free online 5.2
  58. Paul J. Cohen (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences 50 (1963), 1143–1148. free online 1.1
  59. David Cohen-Steiner, Herbert Edelsbrunner and John Harer (2007). Stability of persistence diagrams. Discrete & Computational Geometry 37, 103–120. doi 3.8
  60. Keith Conrad (2025). Quotient groups. expository note, University of Connecticut (undated; version of September 2025). free online 1.4
  61. Peter Constantin, Mihaela Ignatova and Vlad Vicol (2026). Regularity of asymptotically axisymmetric solutions to the 3D Navier–Stokes equations with analytic forcing. preprint. free online 4.3
  62. John H. Conway (1970). An enumeration of knots and links, and some of their algebraic properties. in J. Leech (ed.), Computational Problems in Abstract Algebra, Pergamon, Oxford, 1970, 329–358. doi 3.7
  63. Keenan Crane (2019). Differential forms in ℝⁿ. lecture slides, Discrete Differential Geometry (CMU 15-458/858), Carnegie Mellon University. free online 4.3
  64. Keenan Crane (2025). Discrete Differential Geometry: An Applied Introduction. lecture notes, Carnegie Mellon University (last updated January 2025). free online 4.14.8
  65. Ákos Császár (1949). A polyhedron without diagonals. Acta Scientiarum Mathematicarum (Szeged) 13, 140–142. 2.5
  66. Justin Michael Curry (2014). Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania. free online 4.7
  67. Jonathan J. Danon, Anneke Krüger, David A. Leigh, Jean-François Lemonnier, Alexander J. Stephens, Iñigo J. Vitorica-Yrezabal and Steffen L. Woltering (2017). Braiding a molecular knot with eight crossings. Science 355 (2017), 159–162. doi 3.7
  68. Georges de Rham (1931). Sur l’analysis situs des variétés à n dimensions. Journal de Mathématiques Pures et Appliquées, 9e série, 10, 115–200 (his thesis). free online 4.4
  69. Vin de Silva and Robert Ghrist (2007). Coverage in sensor networks via persistent homology. Algebraic & Geometric Topology 7, 339–358. free online 0.13.84.75.3
  70. Vin de Silva, Dmitriy Morozov and Mikael Vejdemo-Johansson (2011). Persistent cohomology and circular coordinates. Discrete & Computational Geometry 45, 737–759. doi 5.3
  71. Max Dehn (1914). Die beiden Kleeblattschlingen. Mathematische Annalen 75 (1914), 402–413. doi 3.7
  72. Cecil Jose A. Delfinado and Herbert Edelsbrunner (1995). An incremental algorithm for Betti numbers of simplicial complexes on the 3-sphere. Computer Aided Geometric Design 12(7), 771–784. doi 3.3
  73. Pierre Deligne (1974). La conjecture de Weil. I. Publications Mathématiques de l’IHÉS 43, 273–307. free online 5.3
  74. Pierre Deligne (2000). The Hodge conjecture. official problem description, Clay Mathematics Institute. free online 5.3
  75. Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest and Mihai Pătraşcu (2014). Picture-Hanging Puzzles. Theory of Computing Systems 54(4), 531–550. free online 2.3
  76. Reinhard Diestel (2025). Graph Theory (6th edition). Springer, Graduate Texts in Mathematics 173 (free preview of the main text at diestel-graph-theory.com). publisher 3.1
  77. Christiane O. Dietrich-Buchecker and Jean-Pierre Sauvage (1989). A synthetic molecular trefoil knot. Angewandte Chemie International Edition 28 (1989), 189–192. doi 3.7
  78. Paul A. M. Dirac (1931). Quantised singularities in the electromagnetic field. Proceedings of the Royal Society of London A 133, 60–72. doi 4.44.8
  79. Simon K. Donaldson (1983). An application of gauge theory to four-dimensional topology. Journal of Differential Geometry 18(2), 279–315. doi 4.6
  80. Ed Dubinsky, Jennie Dautermann, Uri Leron and Rina Zazkis (1994). On learning fundamental concepts of group theory. Educational Studies in Mathematics 27(3), 267–305. doi 1.21.4
  81. Emilien Dupont, Marvin Eisenberger, Borislav Kozlovskii, Abbas Mehrabian, Francisco J. R. Ruiz, Abigail See, Renfei Zhou, Josh Alman, Virginia Vassilevska Williams and Matej Balog (2026). Improving the matrix multiplication exponent with modern optimization and AlphaEvolve. preprint. free online 3.4
  82. Walther von Dyck (1888). Beiträge zur Analysis situs. Mathematische Annalen 32, 457–512. doi 2.4
  83. Herbert Edelsbrunner, David Letscher and Afra Zomorodian (2002). Topological persistence and simplification. Discrete & Computational Geometry 28, 511–533. doi 3.8
  84. Herbert Edelsbrunner and John Harer (2008). Persistent homology — a survey. in Surveys on Discrete and Computational Geometry: Twenty Years Later, Contemporary Mathematics 453, AMS, 257–282. free online 3.8
  85. Herbert Edelsbrunner and John L. Harer (2010). Computational Topology: An Introduction. American Mathematical Society. doi 2.53.8
  86. Werner Ehrenberg and Raymond E. Siday (1949). The refractive index in electron optics and the principles of dynamics. Proceedings of the Physical Society B 62(1), 8–21. doi 4.4
  87. Samuel Eilenberg and Norman Steenrod (1945). Axiomatic approach to homology theory. Proceedings of the National Academy of Sciences 31(4), 117–120. free online 5.2
  88. Samuel Eilenberg and Norman Steenrod (1952). Foundations of Algebraic Topology. Princeton University Press. doi 3.65.2
  89. Samuel Eilenberg and Saunders Mac Lane (1942a). Group extensions and homology. Annals of Mathematics 43(4), 757–831. doi 5.1
  90. Samuel Eilenberg and Saunders Mac Lane (1942b). Natural isomorphisms in group theory. Proceedings of the National Academy of Sciences 28(12), 537–543. free online 5.1
  91. Samuel Eilenberg and Saunders Mac Lane (1945). General theory of natural equivalences. Transactions of the American Mathematical Society 58, 231–294. doi 5.1
  92. Paul C. Eklof (1976). Whitehead’s problem is undecidable. The American Mathematical Monthly 83 (1976), 775–788. doi 1.4
  93. Leonhard Euler (1741). Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae 8, 128–140 (Euler Archive E53). free online 0.13.1
  94. Leonhard Euler (1758). Elementa doctrinae solidorum. Novi Commentarii academiae scientiarum Petropolitanae 4, 109–140 (Euler Archive E230). free online 0.12.6
  95. Leonhard Euler (1758). Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita. Novi Commentarii academiae scientiarum Petropolitanae 4, 140–160 (Euler Archive E231). free online 2.6
  96. Leonhard Euler (1767). Recherches sur la courbure des surfaces. Mémoires de l’académie des sciences de Berlin 16, 119–143; written 1763 (Euler Archive E333). free online 4.8
  97. Michael Farber (2003). Topological complexity of motion planning. Discrete & Computational Geometry 29, 211–221. free online 0.15.3
  98. P. J. Federico (1982). Descartes on Polyhedra: A Study of the De Solidorum Elementis. Springer, Sources in the History of Mathematics and Physical Sciences 4. doi 0.12.6
  99. Charles L. Fefferman (2000). Existence and smoothness of the Navier–Stokes equation. official problem description, Clay Mathematics Institute. free online 4.3
  100. Michelle Feng and Mason A. Porter (2021). Persistent homology of geospatial data: a case study with voting. SIAM Review 63(1), 67–99. free online 3.8
  101. J. V. Field (1999). Johannes Kepler. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.6
  102. Robert Foote, Mark Levi and Serge Tabachnikov (2013). Tractrices, bicycle tire tracks, hatchet planimeters, and a 100-year-old conjecture. The American Mathematical Monthly 120(3), 199–216. free online 4.3
  103. George K. Francis and Jeffrey R. Weeks (1999). Conway’s ZIP proof. The American Mathematical Monthly 106(5), 393–399. free online 2.4
  104. Fritz Frankl and Lev Pontryagin (1930). Ein Knotensatz mit Anwendung auf die Dimensionstheorie. Mathematische Annalen 102 (1930), 785–789. doi 3.7
  105. Philip Franklin (1934). A six color problem. Journal of Mathematics and Physics 13, 363–369. doi 2.5
  106. Maurice Fréchet (1906). Sur quelques points du calcul fonctionnel. Rendiconti del Circolo Matematico di Palermo 22, 1–72 (his doctoral thesis). doi 2.1
  107. Michael H. Freedman (1982). The topology of four-dimensional manifolds. Journal of Differential Geometry 17(3), 357–453. doi 2.44.6
  108. Michael Freedman, Robert Gompf, Scott Morrison and Kevin Walker (2010). Man and machine thinking about the smooth 4-dimensional Poincaré conjecture. Quantum Topology 1 (2010), 171–208. doi 2.4
  109. Mikio Furuta (2001). Monopole equation and the 11/8-conjecture. Mathematical Research Letters 8 (2001), 279–291. doi 4.6
  110. David E. Galewski and Ronald J. Stern (1980). Classification of simplicial triangulations of topological manifolds. Annals of Mathematics 111 (1980), 1–34. doi 2.5
  111. Jean Gallier and Dianna Xu (2013). A Guide to the Classification Theorem for Compact Surfaces. Springer, Geometry and Computing. doi 2.4
  112. Ronald W. Gatterdam (1981). The planimeter as an example of Green’s theorem. The American Mathematical Monthly 88(9), 701–704. doi 4.3
  113. Carl Friedrich Gauss (1801). Disquisitiones Arithmeticae. Gerhard Fleischer, Leipzig; English translation by Arthur A. Clarke (Yale, 1966; Springer, 1986). free online 1.2
  114. Carl Friedrich Gauss (1827). General Investigations of Curved Surfaces (Disquisitiones generales circa superficies curvas). translated by J. C. Morehead and A. M. Hiltebeitel, Princeton University Library, 1902; Project Gutenberg. free online 4.8
  115. Jim Geelen (2026). OpenAI’s proof of the Cycle Double Cover Theorem. expository preprint. free online 3.1
  116. Robert Ghrist (2008). Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society 45(1), 61–75. free online 3.8
  117. Robert Ghrist and Zoe Cooperband (2025). Obstructions to reality: torsors & visual paradox. preprint. free online 4.1
  118. Lewis Ghrist and Robert Ghrist (2026). Impossible by degrees: cohomology & bistable visual paradox. preprint. free online 4.1
  119. Chad Giusti, Eva Pastalkova, Carina Curto and Vladimir Itskov (2015). Clique topology reveals intrinsic geometric structure in neural correlations. Proceedings of the National Academy of Sciences 112(44), 13455–13460. free online 0.13.85.3
  120. Kurt Gödel (1940). The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. Annals of Mathematics Studies 3, Princeton University Press. 1.1
  121. Cameron McA. Gordon and John Luecke (1989). Knots are determined by their complements. Journal of the American Mathematical Society 2 (1989), 371–415. free online 3.7
  122. Fernando Q. Gouvêa (2011). Was Cantor surprised?. The American Mathematical Monthly 118(3), 198–209. doi 3.5
  123. J. C. Gower and G. J. S. Ross (1969). Minimum spanning trees and single linkage cluster analysis. Journal of the Royal Statistical Society, Series C (Applied Statistics) 18(1), 54–64. doi 3.8
  124. Joseph F. Grcar (2011). How ordinary elimination became Gaussian elimination. Historia Mathematica 38(2), 163–218. free online 1.5
  125. Joshua Evan Greene and Andrew Lobb (2021). The rectangular peg problem. Annals of Mathematics 194 (2021), 509–517. doi 3.5
  126. Alexander Grothendieck (1957). Sur quelques points d’algèbre homologique. Tôhoku Mathematical Journal 9, 119–221. free online 4.75.15.3
  127. Jacques Hadamard (1893). Résolution d’une question relative aux déterminants. Bulletin des Sciences Mathématiques 17 (1893), 240–246. 1.5
  128. Wolfgang Haken (1961). Theorie der Normalflächen: Ein Isotopiekriterium für den Kreisknoten. Acta Mathematica 105 (1961), 245–375. doi 3.7
  129. F. Duncan M. Haldane (1988). Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Physical Review Letters 61, 2015–2018. doi 4.8
  130. Thomas C. Hales (2007). Jordan’s proof of the Jordan curve theorem. Studies in Logic, Grammar and Rhetoric 10(23), 45–60. free online 3.5
  131. Stephen Halperin (1985). Rational homotopy and torus actions. in I. M. James (ed.), Aspects of Topology, London Mathematical Society Lecture Note Series 93, Cambridge University Press, 1985, 293–306. 3.3
  132. Benjamin Halpern and Charles Weaver (1977). Inverting a cylinder through isometric immersions and isometric embeddings. Transactions of the American Mathematical Society 230 (1977), 41–70. free online 2.2
  133. Richard Hammack (2018). Book of Proof (3rd edition). self-published. free online 1.2
  134. Jakob Hansen and Robert Ghrist (2019). Toward a spectral theory of cellular sheaves. Journal of Applied and Computational Topology 3, 315–358. free online 4.7
  135. M. Zahid Hasan and Charles L. Kane (2010). Colloquium: Topological insulators. Reviews of Modern Physics 82, 3045–3067. free online 4.8
  136. C. Brian Haselgrove (1958). A disproof of a conjecture of Pólya. Mathematika 5 (1958), 141–145. doi 0.2
  137. Joel Hass, Jeffrey C. Lagarias and Nicholas Pippenger (1999). The computational complexity of knot and link problems. Journal of the ACM 46 (1999), 185–211. doi 3.7
  138. Allen Hatcher (2002). Algebraic Topology. Cambridge University Press. free online 1.31.42.22.32.42.52.63.13.23.33.53.63.84.14.24.54.64.75.15.25.3
  139. Allen Hatcher (2017). Vector Bundles and K-Theory (version 2.2). book in progress, Cornell University. free online 4.85.3
  140. Jean-Claude Hausmann (1995). On the Vietoris–Rips complexes and a cohomology theory for metric spaces. in Prospects in Topology, Annals of Mathematics Studies 138, Princeton University Press, 175–188. doi 3.8
  141. Jean-Claude Hausmann (2014). Mod Two Homology and Cohomology. Springer, Universitext (a corrected 2022 version is free on the author’s web page). free online 3.2
  142. George Havas, G. E. Wall and J. W. Wamsley (1974). The two generator restricted Burnside group of exponent five. Bulletin of the Australian Mathematical Society 10 (1974), 459–470. free online 1.3
  143. Harald A. Helfgott (2013). The ternary Goldbach conjecture is true. preprint; a book-length version is to appear in Annals of Mathematics Studies. free online 0.2
  144. Harald A. Helfgott (2017). Graph isomorphisms in quasi-polynomial time. English version, with additions, of Séminaire Bourbaki exposé 1125. free online 1.2
  145. Carl Hierholzer and Christian Wiener (1873). Ueber die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Mathematische Annalen 6, 30–32 (published after Hierholzer’s death by Christian Wiener). doi 3.1
  146. Michael A. Hill, Michael J. Hopkins and Douglas C. Ravenel (2016). On the nonexistence of elements of Kervaire invariant one. Annals of Mathematics 184 (2016), 1–262. doi 5.2
  147. Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G. Escolar, Kaname Matsue and Yasumasa Nishiura (2016). Hierarchical structures of amorphous solids characterized by persistent homology. Proceedings of the National Academy of Sciences 113(26), 7035–7040. free online 3.85.3
  148. Friedrich Hirzebruch (1999). Emmy Noether and topology. in M. Teicher (ed.), The Heritage of Emmy Noether, Israel Mathematical Conference Proceedings 12, Bar-Ilan University, 57–65 (lecture of 1996; MPIM preprint 1997-34). free online 3.33.5
  149. W. V. D. Hodge (1950). The topological invariants of algebraic varieties. Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, 1950, vol. 1, 182–192. 5.3
  150. Heinz Hopf (1927). Vektorfelder in n-dimensionalen Mannigfaltigkeiten. Mathematische Annalen 96, 225–249. doi 4.8
  151. Heinz Hopf (1931). Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Mathematische Annalen 104, 637–665. doi 5.3
  152. Heinz Hopf (1935). Über die Drehung der Tangenten und Sehnen ebener Kurven. Compositio Mathematica 2, 50–62. free online 4.8
  153. Michael J. Hopkins, Jianfeng Lin, XiaoLin Danny Shi and Zhouli Xu (2022). Intersection forms of spin 4-manifolds and the Pin(2)-equivariant Mahowald invariant. Communications of the American Mathematical Society 2 (2022), 22–132. free online 4.6
  154. Jim Hoste (2005). The enumeration and classification of knots and links. in W. Menasco and M. Thistlethwaite (eds.), Handbook of Knot Theory, Elsevier, 2005, 209–232. doi 3.7
  155. Jim Hoste, Morwen Thistlethwaite and Jeff Weeks (1998). The first 1,701,936 knots. The Mathematical Intelligencer 20(4) (1998), 33–48. doi 3.7
  156. Michael Hutchings (2011). Cup product and intersections. lecture notes, Math 215B, University of California, Berkeley. free online 4.5
  157. Daniel C. Isaksen, Guozhen Wang and Zhouli Xu (2023). Stable homotopy groups of spheres: from dimension 0 to 90. Publications Mathématiques de l’IHÉS 137, 107–243. free online 5.3
  158. Claudia Weill (1980). It’s My Turn (film, directed by Claudia Weill). transcript of the opening lecture at Burkard Polster and Marty Ross, Mathematics Goes to the Movies. free online 5.2
  159. Xiaoye Jiang, Lek-Heng Lim, Yuan Yao and Yinyu Ye (2011). Statistical ranking and combinatorial Hodge theory. Mathematical Programming 127, 203–244. doi 4.15.3
  160. Dale M. Johnson (1979). The problem of the invariance of dimension in the growth of modern topology, part I. Archive for History of Exact Sciences 20, 97–188. free online 3.5
  161. Vaughan F. R. Jones (1985). A polynomial invariant for knots via von Neumann algebras. Bulletin of the American Mathematical Society 12 (1985), 103–111. free online 3.7
  162. Vaughan F. R. Jones (2000). Ten problems. in Mathematics: Frontiers and Perspectives, American Mathematical Society, 2000, 79–91. 3.7
  163. Thomas W. Judson (2025). Abstract Algebra: Theory and Applications. Annual edition 2025, free under the GNU Free Documentation License. free online 1.21.31.4
  164. Heinrich Jung (1901). Über die kleinste Kugel, die eine räumliche Figur einschliesst. Journal für die reine und angewandte Mathematik 123, 241–257. doi 3.8
  165. Gil Kalai (1989). The number of faces of centrally-symmetric polytopes. Graphs and Combinatorics 5 (1989), 389–391. doi 2.6
  166. Irving Kaplansky (1991). Reminiscences. in J. H. Ewing and F. W. Gehring (eds.), Paul Halmos: Celebrating 50 Years of Mathematics, Springer, 87–89. doi 1.5
  167. Victor J. Katz (1979). The history of Stokes’ theorem. Mathematics Magazine 52(3), 146–156. doi 4.3
  168. Louis H. Kauffman (1987). State models and the Jones polynomial. Topology 26 (1987), 395–407. doi 3.7
  169. J. L. Kelley and Everett Pitcher (1947). Exact homomorphism sequences in homology theory. Annals of Mathematics 48(3), 682–709. doi 3.6
  170. Johannes Kepler (1619). Harmonices Mundi. Linz; Book V translated by Charles Glenn Wallis as Harmonies of the World (1939). 4.6
  171. Mikhail Khovanov (2000). A categorification of the Jones polynomial. Duke Mathematical Journal 101(3), 359–426. doi 3.75.3
  172. Edward C. Kirby, Roger B. Mallion, Paul Pollak and Paweł J. Skrzyński (2016). What Kirchhoff actually did concerning spanning trees in electrical networks and its relationship to modern graph-theoretical work. Croatica Chemica Acta 89(4). free online 3.1
  173. Gustav Kirchhoff (1847). Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Annalen der Physik 148(12), 497–508. doi 2.63.1
  174. Alexei Kitaev (2009). Periodic table for topological insulators and superconductors. AIP Conference Proceedings 1134, 22–30. free online 5.3
  175. Dustin Kleckner and William T. M. Irvine (2013). Creation and dynamics of knotted vortices. Nature Physics 9 (2013), 253–258. doi 3.7
  176. Felix Klein (1882). Über Riemanns Theorie der algebraischen Functionen und ihrer Integrale. Teubner, Leipzig. free online 2.2
  177. Peter B. Kronheimer and Tomasz S. Mrowka (2011). Khovanov homology is an unknot-detector. Publications Mathématiques de l’IHÉS 113, 97–208. free online 3.75.3
  178. Jeremy Kun (2013). Homology Theory — A Primer. Math ∩ Programming (blog), 3 April 2013. free online 3.13.2
  179. Rob Kusner (1987). Conformal geometry and complete minimal surfaces. Bulletin of the American Mathematical Society 17(2), 291–295. free online 2.2
  180. Marc Lackenby (2015). A polynomial upper bound on Reidemeister moves. Annals of Mathematics 182 (2015), 491–564. doi 3.7
  181. Marc Lackenby (2021). The efficient certification of knottedness and Thurston norm. Advances in Mathematics 387 (2021), 107796. doi 3.7
  182. Imre Lakatos (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press (edited by John Worrall and Elie Zahar). doi 2.6
  183. Lam Lay Yong and Ang Tian Se (2004). Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China (revised edition). World Scientific; includes a complete English translation of the Sunzi suanjing. doi 1.4
  184. Evelyn Lamb (2017). What does compactness really mean?. Roots of Unity, Scientific American blog, 25 May 2017. free online 2.1
  185. Leon J. Lander and Thomas R. Parkin (1966). Counterexample to Euler’s conjecture on sums of like powers. Bulletin of the American Mathematical Society 72 (1966), 1079. free online 0.2
  186. John M. Lee (2011). Introduction to Topological Manifolds (2nd edition). Springer, Graduate Texts in Mathematics 202. doi 2.4
  187. John M. Lee (2013). Introduction to Smooth Manifolds (2nd edition). Springer, Graduate Texts in Mathematics 218. doi 2.44.34.4
  188. Solomon Lefschetz (1930). Topology. American Mathematical Society, Colloquium Publications 12. doi 3.5
  189. Tom Leinster (2014). Basic Category Theory. Cambridge University Press. free online 5.1
  190. Jean Leray (1934). Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Mathematica 63 (1934), 193–248. doi 4.3
  191. Bernadette Lessel (2023). Walther Mayer – more than “Einstein’s calculator”. Bits of History, Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, 22 March 2023. free online 3.6
  192. Simon Lhuilier (1813). Mémoire sur la polyèdrométrie ; contenant une démonstration directe du théorème d’Euler sur les polyèdres, et un examen des diverses exceptions auxquelles ce théorème est assujetti. Annales de mathématiques pures et appliquées 3 (1812–1813), 169–189. free online 0.12.6
  193. Lek-Heng Lim (2020). Hodge Laplacians on graphs. SIAM Review 62(3), 685–715. free online 4.14.45.3
  194. Weinan Lin, Guozhen Wang and Zhouli Xu (2024). On the last Kervaire invariant problem. preprint. free online 5.2
  195. Leroy F. Liu, Roxanne E. Depew and James C. Wang (1976). Knotted single-stranded DNA rings: a novel topological isomer of circular single-stranded DNA formed by treatment with Escherichia coli ω protein. Journal of Molecular Biology 106 (1976), 439–452. doi 3.7
  196. Frank H. Lutz (2008). Enumeration and random realization of triangulated surfaces. in A. I. Bobenko et al. (eds.), Discrete Differential Geometry, Oberwolfach Seminars 38, Birkhäuser, 235–253. free online 2.5
  197. Saunders Mac Lane (1971). Categories for the Working Mathematician. Springer, Graduate Texts in Mathematics 5 (2nd edition 1998). doi 5.1
  198. Duarte Maia (2025). On the visualization of differential forms. notes, University of Chicago (a transcription of two 2021 blog posts, based on an informal note by Dan Piponi). free online 4.3
  199. Ciprian Manolescu (2016). Pin(2)-equivariant Seiberg–Witten Floer homology and the Triangulation Conjecture. Journal of the American Mathematical Society 29, 147–176. free online 2.5
  200. Eyal Markman (2025). Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds. preprint. free online 5.3
  201. A. A. Markov (1936). Über die freie Äquivalenz der geschlossenen Zöpfe. Recueil Mathématique (Matematicheskii Sbornik) 1(43) (1936), 73–78. 3.7
  202. A. A. Markov (1958). The insolubility of the problem of homeomorphy. Doklady Akademii Nauk SSSR 121(2), 218–220 (in Russian). 2.4
  203. William S. Massey (1998). Higher order linking numbers. Journal of Knot Theory and Its Ramifications 7 (1998), 393–414; first published in the proceedings of a 1968 conference on algebraic topology. doi 4.5
  204. “Josh” (2011). Answer to “Intuition of the meaning of homology groups”. Mathematics Stack Exchange, question 40149, answer 40151 (19 May 2011). free online 3.1
  205. Benjamin Matschke (2014). A survey on the square peg problem. Notices of the American Mathematical Society 61 (2014), 346–352. free online 3.5
  206. Yukio Matsumoto (1982). On the bounding genus of homology 3-spheres. Journal of the Faculty of Science, University of Tokyo, Section IA, Mathematics 29 (1982), 287–318. 4.6
  207. James Clerk Maxwell (1873). A Treatise on Electricity and Magnetism, vol. 1. Clarendon Press, Oxford. free online 4.3
  208. Walther Mayer (1929). Über abstrakte Topologie. Monatshefte für Mathematik und Physik 36, 1–42. doi 3.23.6
  209. Colin McLarty (2007). The rising sea: Grothendieck on simplicity and generality. in J. J. Gray and K. H. Parshall (eds), Episodes in the History of Modern Algebra (1800–1950), AMS/LMS History of Mathematics 32 (preprint, 2003). free online 5.25.3
  210. Haynes Miller (2000). Leray in Oflag XVIIA: the origins of sheaf theory, sheaf cohomology, and spectral sequences. Gazette des Mathématiciens 84 (supplement), 17–34. free online 4.75.2
  211. John Milnor (1956). On manifolds homeomorphic to the 7-sphere. Annals of Mathematics 64(2), 399–405. doi 2.4
  212. John Milnor (1961). Two complexes which are homeomorphic but combinatorially distinct. Annals of Mathematics 74(3), 575–590. doi 2.6
  213. John Milnor (1962). On axiomatic homology theory. Pacific Journal of Mathematics 12(1), 337–341. free online 5.2
  214. John W. Milnor (1965). Topology from the Differentiable Viewpoint. University Press of Virginia (reprinted by Princeton University Press, 1997). publisher 3.54.8
  215. John W. Milnor and James D. Stasheff (1974). Characteristic Classes. Princeton University Press, Annals of Mathematics Studies 76. doi 4.8
  216. Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (1973). Gravitation. W. H. Freeman (reprinted by Princeton University Press, 2017). publisher 4.3
  217. August Ferdinand Möbius (1827). Der barycentrische Calcul. Johann Ambrosius Barth, Leipzig. free online 2.5
  218. Gregory H. Moore (2008). The emergence of open sets, closed sets, and limit points in analysis and topology. Historia Mathematica 35(3), 220–241. doi 2.1
  219. John Morgan and Gang Tian (2007). Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs 3, American Mathematical Society. free online 2.4
  220. James R. Munkres (1984). Elements of Algebraic Topology. Addison-Wesley (reprinted by CRC Press, 2018). doi 3.33.4
  221. James R. Munkres (2000). Topology (2nd edition). Prentice Hall. 1.22.12.2
  222. Kunio Murasugi (1987). Jones polynomials and classical conjectures in knot theory. Topology 26 (1987), 187–194. doi 3.7
  223. John Nash (1951). Non-cooperative games. Annals of Mathematics 54(2), 286–295. doi 0.1
  224. Tristan Needham (2021). Visual Differential Geometry and Forms. Princeton University Press. doi 4.8
  225. Peter M. Neumann (2011). The Mathematical Writings of Évariste Galois. European Mathematical Society, Heritage of European Mathematics. doi 1.3
  226. Monica Nicolau, Arnold J. Levine and Gunnar Carlsson (2011). Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proceedings of the National Academy of Sciences 108(17), 7265–7270. free online 3.8
  227. The Royal Swedish Academy of Sciences (2016). The Nobel Prize in Physics 2016. NobelPrize.org. free online 0.14.8
  228. John J. O’Connor and Edmund F. Robertson (1998). Julius Wilhelm Richard Dedekind. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.5
  229. John J. O’Connor and Edmund F. Robertson (1999). Andrey Nikolaevich Kolmogorov. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.2
  230. John J. O’Connor and Edmund F. Robertson (1999). Pavel Sergeevich Aleksandrov. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.7
  231. John J. O’Connor and Edmund F. Robertson (2000). Samuel Eilenberg. MacTutor History of Mathematics Archive, University of St Andrews (quotes H. Bass on his art collection). free online 5.1
  232. John J. O’Connor and Edmund F. Robertson (2000). James Waddell Alexander. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.2
  233. John J. O’Connor and Edmund F. Robertson (2000). Norman Earl Steenrod. MacTutor History of Mathematics Archive, University of St Andrews. free online 5.2
  234. John J. O’Connor and Edmund F. Robertson (2000). Heinz Hopf. MacTutor History of Mathematics Archive, University of St Andrews. free online 5.3
  235. John J. O’Connor and Edmund F. Robertson (2000). Emil Artin. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.7
  236. John J. O’Connor and Edmund F. Robertson (2000). Herbert Seifert. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.7
  237. John J. O’Connor and Edmund F. Robertson (2000). Eduard Čech. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.7
  238. John J. O’Connor and Edmund F. Robertson (2001). Marie Ennemond Camille Jordan. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.5
  239. John J. O’Connor and Edmund F. Robertson (2001). Jean Leray. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.7
  240. John J. O’Connor and Edmund F. Robertson (2002). Gustav Robert Kirchhoff. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.1
  241. John J. O’Connor and Edmund F. Robertson (2003). Paul Adrien Maurice Dirac. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.8
  242. John J. O’Connor and Edmund F. Robertson (2003). William Thomson (Lord Kelvin). MacTutor History of Mathematics Archive, University of St Andrews. free online 4.3
  243. John J. O’Connor and Edmund F. Robertson (2003). George Gabriel Stokes. MacTutor History of Mathematics Archive, University of St Andrews (quotes his letters of 1857 to Mary Susanna Robinson). free online 4.3
  244. John J. O’Connor and Edmund F. Robertson (2003). Henri Poincaré. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.3
  245. John J. O’Connor and Edmund F. Robertson (2003). Luitzen Egbertus Jan Brouwer. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.5
  246. John J. O’Connor and Edmund F. Robertson (2003). Solomon Lefschetz. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.5
  247. John J. O’Connor and Edmund F. Robertson (2003). Nine Chapters on the Mathematical Art. MacTutor History of Mathematics Archive, University of St Andrews. free online 1.5
  248. John J. O’Connor and Edmund F. Robertson (2004). John Horton Conway. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.7
  249. John J. O’Connor and Edmund F. Robertson (2004). Henri Paul Cartan. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.7
  250. John J. O’Connor and Edmund F. Robertson (2005). Henry John Stephen Smith. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.4
  251. John J. O’Connor and Edmund F. Robertson (2005). Hassler Whitney. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.5
  252. John J. O’Connor and Edmund F. Robertson (2006). Raoul Bott. MacTutor History of Mathematics Archive, University of St Andrews (with E. F. Robertson’s own memory of Bott at Warwick). free online 5.3
  253. John J. O’Connor and Edmund F. Robertson (2008). Eduard L. Stiefel. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.8
  254. John J. O’Connor and Edmund F. Robertson (2009). Friedrich Ernst Peter Hirzebruch. MacTutor History of Mathematics Archive, University of St Andrews. free online 5.3
  255. John J. O’Connor and Edmund F. Robertson (2009). Vaughan Frederick Randal Jones. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.7
  256. John J. O’Connor and Edmund F. Robertson (2014). Alexander Grothendieck. MacTutor History of Mathematics Archive, University of St Andrews (quotes Grothendieck’s account of his 1967 visit to Vietnam). free online 5.1
  257. John J. O’Connor and Edmund F. Robertson (2014). Shiing-shen Chern. MacTutor History of Mathematics Archive, University of St Andrews (quotes Chern’s interviews). free online 4.8
  258. John J. O’Connor and Edmund F. Robertson (2014). Élie Joseph Cartan. MacTutor History of Mathematics Archive, University of St Andrews. free online 4.4
  259. John J. O’Connor and Edmund F. Robertson (2014). Emmy Amalie Noether. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.3
  260. John J. O’Connor and Edmund F. Robertson (2014). Witold Hurewicz. MacTutor History of Mathematics Archive, University of St Andrews (quotes H. Freudenthal and S. Eilenberg). free online 5.3
  261. John J. O’Connor and Edmund F. Robertson (2014). Michael Francis Atiyah. MacTutor History of Mathematics Archive, University of St Andrews. free online 5.3
  262. John J. O’Connor and Edmund F. Robertson (2014). Kurt Werner Friedrich Reidemeister. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.7
  263. John J. O’Connor and Edmund F. Robertson (2015). Leslie Saunders Mac Lane. MacTutor History of Mathematics Archive, University of St Andrews. free online 5.1
  264. John J. O’Connor and Edmund F. Robertson (2015). Georges de Rham. MacTutor History of Mathematics Archive, University of St Andrews (quotes R. Bott, Notices of the AMS 38 (1991), 114–115). free online 4.4
  265. John J. O’Connor and Edmund F. Robertson (2015). Leopold Vietoris. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.6
  266. John J. O’Connor and Edmund F. Robertson (2015). Enrico Betti. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.3
  267. John J. O’Connor, Edmund F. Robertson and H. J. Munkholm (2010). Poul Heegaard. MacTutor History of Mathematics Archive, University of St Andrews. free online 3.2
  268. Bruno Vallet and others (2009). A159257: Rank deficiency of the Lights Out problem of size n. The On-Line Encyclopedia of Integer Sequences. free online 1.5
  269. The OEIS Foundation (2026). Sequence A007299: Hadamard matrices of order 4n. The On-Line Encyclopedia of Integer Sequences, with links to the matrices of orders 668 to 1964 found in August 2026. free online 1.5
  270. Tomás Oliveira e Silva, Siegfried Herzog and Silvio Pardi (2014). Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10¹⁸. Mathematics of Computation 83 (2014), 2033–2060. doi 0.2
  271. OpenAI (2026). An upper bound of 9/4 for the matrix multiplication exponent. manuscript produced by an AI model, in the repository github.com/openai/math (family 107), October 2026. free online 3.4
  272. OpenAI (2026). A proof of the cycle double cover conjecture. three-page note, July 2026; the proof is credited to the model GPT-5.6 Sol Ultra. free online 3.1
  273. Nina Otter, Mason A. Porter, Ulrike Tillmann, Peter Grindrod and Heather A. Harrington (2017). A roadmap for the computation of persistent homology. EPJ Data Science 6, 17. free online 3.8
  274. Peter Ozsváth and Zoltán Szabó (2004). Holomorphic disks and knot invariants. Advances in Mathematics 186 (2004), 58–116. doi 5.3
  275. Peter Ozsváth and Zoltán Szabó (2004). Holomorphic disks and genus bounds. Geometry & Topology 8 (2004), 311–334. free online 5.3
  276. Raymond E. A. C. Paley (1933). On orthogonal matrices. Journal of Mathematics and Physics 12 (1933), 311–320. doi 1.5
  277. Giuseppe Peano (1890). Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen 36, 157–160. doi 2.43.5
  278. Roger Penrose and Lionel S. Penrose (1955–59). First models of the “impossible triangle” and “impossible staircase” (birch plywood). Science Museum Group Collection, London, object 1982-899. free online 4.1
  279. Lionel S. Penrose and Roger Penrose (1958). Impossible objects: a special type of visual illusion. British Journal of Psychology 49(1), 31–33. doi 0.14.14.44.7
  280. Roger Penrose (1992). On the cohomology of impossible figures. Leonardo 25(3/4), 245–247. doi 0.14.14.7
  281. Grisha Perelman (2002). The entropy formula for the Ricci flow and its geometric applications. preprint. free online 2.4
  282. Grisha Perelman (2003). Ricci flow with surgery on three-manifolds. preprint. free online 2.4
  283. Kenneth A. Perko (1974). On the classification of knots. Proceedings of the American Mathematical Society 45 (1974), 262–266. free online 3.7
  284. Henri Poincaré (1885). Sur les courbes définies par les équations différentielles (troisième partie). Journal de Mathématiques Pures et Appliquées, 4e série, 1, 167–244. free online 4.8
  285. Henri Poincaré (1895). Analysis situs. Journal de l’École Polytechnique, 2e série, cahier 1; English translation by John Stillwell in Papers on Topology (AMS, 2010). doi 0.12.33.23.34.65.3
  286. Henri Poincaré (1899). Complément à l’Analysis situs. Rendiconti del Circolo Matematico di Palermo 13, 285–343; English translation by John Stillwell in Papers on Topology (AMS, 2010). doi 3.23.34.6
  287. Henri Poincaré (1900). Second complément à l’Analysis Situs. Proceedings of the London Mathematical Society 32, 277–308; English translation by John Stillwell in Papers on Topology (AMS, 2010). doi 1.4
  288. Henri Poincaré (1904). Cinquième complément à l’Analysis situs. Rendiconti del Circolo Matematico di Palermo 18, 45–110. doi 2.43.5
  289. Henri Poincaré (1908). Science et méthode. Flammarion, Paris; English translation by George Bruce Halsted as “Science and Method” in The Foundations of Science (Science Press, 1913). free online 0.21.21.3
  290. Henri Poincaré (1914). Science and Method. translated by Francis Maitland, Thomas Nelson and Sons, London (French original: Science et méthode, Flammarion, 1908). free online 5.1
  291. George Pólya (1954). Mathematics and Plausible Reasoning, Volume I: Induction and Analogy in Mathematics. Princeton University Press. 0.2
  292. Pratyush Pranav, Herbert Edelsbrunner, Rien van de Weygaert, Gert Vegter, Michael Kerber, Bernard J. T. Jones and Mathijs Wintraecken (2017). The topology of the cosmic web in terms of persistent Betti numbers. Monthly Notices of the Royal Astronomical Society 465(4), 4281–4310. free online 3.8
  293. Józef H. Przytycki (1998). 3-coloring and other elementary invariants of knots. Banach Center Publications 42 (1998), 275–295. free online 3.7
  294. Tibor Radó (1925). Über den Begriff der Riemannschen Fläche. Acta Scientiarum Mathematicarum (Szeged) 2, 101–121. 2.52.6
  295. Jacob Rasmussen (2010). Khovanov homology and the slice genus. Inventiones Mathematicae 182 (2010), 419–447. doi 5.3
  296. Kurt Reidemeister (1927). Elementare Begründung der Knotentheorie. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5 (1927), 24–32. doi 3.7
  297. Kurt Reidemeister (1932). Knotentheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin. 3.7
  298. Heinrich Reitberger (2002). Leopold Vietoris (1891–2002). Notices of the American Mathematical Society 49(10), 1232–1236. free online 3.63.8
  299. Renzo L. Ricca and Bernardo Nipoti (2011). Gauss’ linking number revisited. Journal of Knot Theory and Its Ramifications 20(10), 1325–1343. doi 3.74.6
  300. David S. Richeson (2008). Euler’s Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press. doi 2.64.8
  301. David S. Richeson (2021). Topology 101: The Hole Truth. Quanta Magazine, 26 January 2021. free online 0.13.1
  302. Bernhard Riemann (1851). Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Inauguraldissertation, Göttingen (transcribed by D. R. Wilkins, Trinity College Dublin). free online 4.7
  303. Bernhard Riemann (1854). On the Hypotheses which lie at the Bases of Geometry. Habilitation lecture, Göttingen, 1854, published 1868; English translation by W. K. Clifford, Nature 8 (1873). free online 0.22.4
  304. Bernhard Riemann (1857). Theorie der Abel’schen Functionen. Journal für die reine und angewandte Mathematik 54, 115–155. doi 3.1
  305. Bernhard Riemann (1859). Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Königlichen Preussischen Akademie der Wissenschaften zu Berlin, November 1859, 671–680. 0.2
  306. Dale Rolfsen (1976). Knots and Links. Publish or Perish, Berkeley; reprinted by AMS Chelsea, 2003. 2.33.7
  307. Richard L. Roth (2001). A history of Lagrange’s theorem on groups. Mathematics Magazine 74(2), 99–108. doi 1.4
  308. Raman Sanyal, Axel Werner and Günter M. Ziegler (2009). On Kalai’s conjectures concerning centrally symmetric polytopes. Discrete & Computational Geometry 41 (2009), 183–198. doi 2.6
  309. Horst Schubert (1949). Die eindeutige Zerlegbarkeit eines Knotens in Primknoten. Sitzungsberichte der Heidelberger Akademie der Wissenschaften, Math.-nat. Klasse, 1949, 3. Abhandlung, 57–104. doi 3.7
  310. Richard Evan Schwartz (2025). The optimal paper Moebius band. Annals of Mathematics 201(1), 291–305. free online 2.2
  311. Herbert Seifert (1935). Über das Geschlecht von Knoten. Mathematische Annalen 110, 571–592. doi 3.74.6
  312. Herbert Seifert and William Threlfall (1934). Lehrbuch der Topologie. Teubner, Leipzig. 3.7
  313. Jean-Pierre Serre (1951). Homologie singulière des espaces fibrés. Applications. Annals of Mathematics 54(3), 425–505. doi 5.2
  314. Jean-Pierre Serre (1953a). Groupes d’homotopie et classes de groupes abéliens. Annals of Mathematics 58(2), 258–294. doi 5.2
  315. Jean-Pierre Serre (1953b). Cohomologie modulo 2 des complexes d’Eilenberg–MacLane. Commentarii Mathematici Helvetici 27, 198–232. doi 5.3
  316. Jean-Pierre Serre (1955). Faisceaux algébriques cohérents. Annals of Mathematics 61(2), 197–278. doi 4.75.3
  317. Paul D. Seymour (1979). Sums of circuits. in J. A. Bondy and U. S. R. Murty (eds.), Graph Theory and Related Topics, Academic Press, 1979, 341–355. 3.1
  318. Ali Shehper, Anibal M. Medina-Mardones, Lucas Fagan, Bartłomiej Lewandowski, Angus Gruen, Yang Qiu, Piotr Kucharski, Zhenghan Wang and Sergei Gukov (2024). What makes math problems hard for reinforcement learning: a case study. preprint. free online 2.3
  319. Saharon Shelah (1974). Infinite abelian groups, Whitehead problem and some constructions. Israel Journal of Mathematics 18 (1974), 243–256. doi 1.4
  320. Daniel Siebert and Steven R. Williams (2003). Students’ understanding of ℤn. Proceedings of the 2003 joint meeting of PME and PME-NA, vol. 4, 167–173. free online 1.4
  321. Barry Simon (1983). Holonomy, the quantum adiabatic theorem, and Berry’s phase. Physical Review Letters 51, 2167–2170. doi 4.8
  322. Henry John Stephen Smith (1861). On systems of linear indeterminate equations and congruences. Philosophical Transactions of the Royal Society of London 151, 293–326. doi 1.41.53.4
  323. Michael Spivak (1965). Calculus on Manifolds. W. A. Benjamin (reprinted by CRC Press, 2018). doi 4.34.4
  324. The Stacks Project Authors (2026). The Stacks Project. an open online reference for algebraic geometry, Columbia University. free online 4.7
  325. John Stillwell (2010). Translator’s introduction, in Henri Poincaré, Papers on Topology: Analysis Situs and Its Five Supplements. American Mathematical Society and London Mathematical Society, History of Mathematics 37. free online 1.4
  326. George Gabriel Stokes (1854). Smith’s Prize examination paper, February 1854. University of Cambridge; question 8 is the theorem now named after Stokes (transcription by the James Clerk Maxwell Foundation). free online 4.3
  327. Gilbert Strang (1993). The fundamental theorem of linear algebra. The American Mathematical Monthly 100(9), 848–855. doi 1.5
  328. Volker Strassen (1969). Gaussian elimination is not optimal. Numerische Mathematik 13 (1969), 354–356. doi 3.4
  329. Peter Guthrie Tait (1877). On knots. Transactions of the Royal Society of Edinburgh 28 (1877), 145–190; reprinted in Scientific Papers, vol. I, Cambridge University Press, 1898, 273–317. doi 3.7
  330. Minoru Tanaka (1980). A numerical investigation on cumulative sum of the Liouville function. Tokyo Journal of Mathematics 3 (1980), 187–189. doi 0.2
  331. Terence Tao (2017). An integration approach to the Toeplitz square peg problem. Forum of Mathematics, Sigma 5 (2017), e30. free online 3.5
  332. William R. Taylor (2000). A deeply knotted protein structure and how it might fold. Nature 406 (2000), 916–919. doi 3.7
  333. Richard Taylor and Andrew Wiles (1995). Ring-theoretic properties of certain Hecke algebras. Annals of Mathematics 141 (1995), 553–572. doi 0.2
  334. Morwen B. Thistlethwaite (1987). A spanning tree expansion of the Jones polynomial. Topology 26 (1987), 297–309. doi 3.7
  335. Morwen B. Thistlethwaite (2001). Links with trivial Jones polynomial. Journal of Knot Theory and Its Ramifications 10 (2001), 641–643. doi 3.7
  336. Morwen B. Thistlethwaite (2025). The enumeration and classification of prime 20-crossing knots. Algebraic & Geometric Topology 25 (2025), 329–344. doi 3.7
  337. René Thom (1954). Quelques propriétés globales des variétés différentiables. Commentarii Mathematici Helvetici 28, 17–86. doi 5.3
  338. William Thomson (1867). On vortex atoms. Proceedings of the Royal Society of Edinburgh 6 (1867), 94–105; also Philosophical Magazine 34 (1867), 15–24. doi 3.7
  339. David J. Thouless, Mahito Kohmoto, M. Peter Nightingale and Marcel den Nijs (1982). Quantized Hall conductance in a two-dimensional periodic potential. Physical Review Letters 49, 405–408. doi 0.14.85.3
  340. Heinrich Tietze (1908). Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten. Monatshefte für Mathematik und Physik 19, 1–118. doi 3.2
  341. Akira Tonomura, Nobuyuki Osakabe, Tsuyoshi Matsuda, Takeshi Kawasaki, Junji Endo, Shinichiro Yano and Hiroji Yamada (1986). Evidence for Aharonov–Bohm effect with magnetic field completely shielded from electron wave. Physical Review Letters 56, 792–795. doi 4.4
  342. Daniel Tubbenhauer (2021). What is…cellular homology? Or: Winding around (slides). Slides for the VisualMath video series on algebraic topology. free online 3.4
  343. Daniel Tubbenhauer (2021). What is…homology intuitively?. VisualMath, YouTube (slides at dtubbenhauer.com). free online 0.13.13.3
  344. Robert E. Tuzun and Adam S. Sikora (2021). Verification of the Jones unknot conjecture up to 24 crossings. Journal of Knot Theory and Its Ramifications 30 (2021), 2150020. doi 3.7
  345. Ravi Vakil (2008). Spectral sequences: friend or foe?. lecture notes, Stanford University (Math 216). free online 5.2
  346. Dirk van Dalen (2013). L. E. J. Brouwer – Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life. Springer. doi 3.5
  347. Oswald Veblen (1905). Theory on plane curves in non-metrical analysis situs. Transactions of the American Mathematical Society 6(1), 83–98. doi 3.5
  348. Oswald Veblen and J. W. Alexander (1913). Manifolds of n dimensions. Annals of Mathematics (2) 14, 163–178 (volume dated 1912–13). doi 3.2
  349. Jeff Viaclovsky (2026). A two-parameter family of complex structures on S⁶. preprint. free online 4.8
  350. Steven Vickers (1989). Topology via Logic. Cambridge University Press, Cambridge Tracts in Theoretical Computer Science 5. 2.1
  351. Leopold Vietoris (1927). Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen. Mathematische Annalen 97, 454–472. doi 3.63.8
  352. Leopold Vietoris (1930). Über die Homologiegruppen der Vereinigung zweier Komplexe. Monatshefte für Mathematik und Physik 37, 159–162. doi 3.6
  353. Klaus von Klitzing, Gerhard Dorda and Michael Pepper (1980). New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters 45, 494–497. doi 4.8
  354. Karl Georg Christian von Staudt (1847). Geometrie der Lage. Bauer und Raspe, Nürnberg. free online 2.6
  355. Mark E. Walker (2017). Total Betti numbers of modules of finite projective dimension. Annals of Mathematics 186 (2017), 641–646. doi 3.3
  356. Frank W. Warner (1983). Foundations of Differentiable Manifolds and Lie Groups. Springer, Graduate Texts in Mathematics 94. doi 5.3
  357. Steven A. Wasserman and Nicholas R. Cozzarelli (1986). Biochemical topology: applications to DNA recombination and replication. Science 232 (1986), 951–960. doi 3.7
  358. Steven A. Wasserman, Jan M. Dungan and Nicholas R. Cozzarelli (1985). Discovery of a predicted DNA knot substantiates a model for site-specific recombination. Science 229 (1985), 171–174. doi 3.7
  359. Charles A. Weibel (1994). An Introduction to Homological Algebra. Cambridge University Press. doi 5.2
  360. Charles A. Weibel (1999). History of homological algebra. in I. M. James (ed.), History of Topology, Elsevier, 797–836. 0.13.13.23.33.64.24.44.55.15.2
  361. Shmuel Weinberger (2011). What is … persistent homology?. Notices of the American Mathematical Society 58(1), 36–39. free online 3.8
  362. Eric W. Weisstein (2026). Hole. MathWorld — A Wolfram Resource (accessed October 2026). free online 0.1
  363. H. Wendt (1937). Die gordische Auflösung von Knoten. Mathematische Zeitschrift 42 (1937), 680–696. 3.7
  364. Hermann Weyl (1913). Die Idee der Riemannschen Fläche. Teubner, Leipzig. free online 2.4
  365. Hermann Weyl (1939). Invariants. Duke Mathematical Journal 5(3), 489–502. doi 2.5
  366. J. H. C. Whitehead (1949). Combinatorial homotopy. I. Bulletin of the American Mathematical Society 55(3), 213–245. free online 2.53.6
  367. Hassler Whitney (1938). On products in a complex. Annals of Mathematics 39(2), 397–432. doi 4.14.24.54.6
  368. Hassler Whitney (1944a). The self-intersections of a smooth n-manifold in 2n-space. Annals of Mathematics 45(2), 220–246. doi 2.4
  369. Hassler Whitney (1944b). The singularities of a smooth n-manifold in (2n − 1)-space. Annals of Mathematics 45(2), 247–293. doi 2.4
  370. Andrew Wiles (1995). Modular elliptic curves and Fermat’s Last Theorem. Annals of Mathematics 141 (1995), 443–551. doi 0.2
  371. Edward Witten (1989). Quantum field theory and the Jones polynomial. Communications in Mathematical Physics 121 (1989), 351–399. doi 3.7
  372. Tai Tsun Wu and Chen Ning Yang (1975). Concept of nonintegrable phase factors and global formulation of gauge fields. Physical Review D 12, 3845–3857. doi 4.44.8
  373. Kelin Xia and Guo-Wei Wei (2014). Persistent homology analysis of protein structure, flexibility, and folding. International Journal for Numerical Methods in Biomedical Engineering 30(8), 814–844. doi 0.13.8
  374. Chris Yu, Henrik Schumacher and Keenan Crane (2021). Repulsive curves. ACM Transactions on Graphics 40:2 (2021), 1–21. doi 3.7
  375. E. C. Zeeman (1963). On the dunce hat. Topology 2(4), 341–358. doi 3.6
  376. Leopold Zoller (2024). The toral rank conjecture. lecture notes for a mini-course, University of Copenhagen, June 2024. free online 3.3
  377. Afra Zomorodian and Gunnar Carlsson (2005). Computing persistent homology. Discrete & Computational Geometry 33, 249–274. doi 3.8

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