Definitions I.17–I.18, printed pp. 8–9, describe an abstract simplicial complex and its topological realization by gluing labelled faces of standard simplexes.
Cited Book I on geometrical operations with a unit segment and Book II on curves. The linked transcription is a modern reproduction, not a claim to be a new printing of the 1637 edition.
This MacTutor biography gives approximate dates and discusses the surviving evidence for Diocles' work on the cissoid of Diocles and the parabola. The footer identifies John Joseph O'Connor and Edmund Frederick Robertson as authors and April 1999 as its revision date.
Cited passages: Romyar Sharifi, Point-Set Topology, Corollary 3.1.10, Proposition 3.1.11, Definition 3.1.12 and Remark 3.1.16: maximal connected subsets and equivalence classes; Romyar Sharifi, Point-Set Topology, Definition 3.1.1 and Example 3.1.3: connectedness and discrete spaces; Romyar Sharifi, Point-Set Topology, Lemma 3.1.13 and Example 3.1.14: closed components and a component which is not open.
Cited passages: Definition 1.1.9: finite complement topology; Definition 1.5.5, Proposition 1.5.11 and Proposition 1.6.20: neighbourhood convergence, uniqueness and the metric formulation; Example 1.5.10 and Proposition 1.5.11: closed points, overlapping neighbourhoods, non-repeating sequences and uniqueness in Hausdorff spaces.
Cited passages: Tom Leinster, General Topology, Example C3.6, p. 78, and the discussion after Lemma C4.12, p. 84: the sine-curve obstruction and nonclosed path components; here the positive graph is restricted to ; Tom Leinster, General Topology, Lemma C1.8, p. 70: a continuous image is connected; Tom Leinster, General Topology, Lemma C4.12 and the following definition, p. 84: path equivalence and maximal path-connected subsets; Tom Leinster, General Topology, Remark C1.2, p. 68, and Proposition C2.6, p. 73: the nonempty convention and the interval characterization.
By Jiří Matoušek. Subtitle: Lectures on Topological Methods in Combinatorics and Geometry. First edition, Springer Berlin, Heidelberg. The publisher dates the printed edition to 2003 and the electronic edition to 2008.
Cited passage: §2.1, p. 22, the antipodal-temperature example and its continuity assumption. The publisher record verifies the book metadata and lists the Borsuk–Ulam chapter on pp. 21–46; the whole book is subscription content.
By Karol Borsuk. Published in Fundamenta Mathematicae, volume 20, pp. 177–190.
Collection of sources; individual chapter authors and dates are recorded below.
Cited passages: Jonny Evans, 3.01 Quotient topology, endpoint identification and circle preimages, notes at 7.39–11.13.
Cited passages: Jonny Evans, 3.02 Quotient topology: continuous maps, notes at 0.14–1.47 and 4.30–7.33.
Collection of sources; individual chapter authors and dates are recorded below.
Publisher: Elsevier B.V..
Cited passages: Symplectic Geometry, §1.1, pp. 82–83: paired bases and Lagrangian spans; §1.5, pp. 90–91: Darboux coordinates; §1.1, p. 83: e-then-f basis and its block matrix; the coordinate formula follows by bilinearity.
Publisher: Wiley Periodicals, Inc.
Published in Random Structures & Algorithms.
Cited passages: An elementary proof of a theorem of Johnson and Lindenstrauss, Theorem 2.1, p. 61: bounds for squared distances.
Linked version dated 1 January 2013.
Cited passages: Jiří Matoušek, Lecture notes on metric embeddings, preface and Section 1.1; Lecture notes on metric embeddings, §2.1(b), p. 21, and Lemma 2.3.1, p. 24: independent Gaussian coefficients and normalization; Lecture notes on metric embeddings, §2.1, pp. 20–21, and proof of Lemma 2.3.1, pp. 25–26: fixed-vector concentration and union bound over pair differences; Matoušek, Section 1.1: distortion and Lipschitz maps.
Publisher: Cambridge University Press.
The inspected PDF carries copyright 2002; the publisher front matter separately records first publication in 2001 and a fourteenth printing in 2015.
Cited passages: Algebraic Topology, §2.2, p. 146, definition of Euler characteristic and Theorem 2.44; Algebraic Topology, §2.2, p. 146, homotopy-type independence immediately before Theorem 2.44.
Additional cited passages: §2.2, Example 2.42, p. 144; Euler characteristic and Theorem 2.44, pp. 146–147. These are chapter locators in Algebraic Topology by Allen Hatcher.
Publisher: University of Chicago Press.
Cited passages: A Concise Course in Algebraic Topology, Chapter 13, §5, p. 103, Example (i), the torus cellular chains and integral homology.
Cited passages: Lecture 5, Definition 5.16, printed p. 19: total branching index as the sum of fibre deficits; Lecture 5, Theorems 5.1 and 5.3; Definition 5.16 and Theorem 5.17, printed pp. 16-19.
Cited passages: Section 3.2, Theorem 3.2 and its triangulation proof, printed pp. 38-39; Section 3.3, printed pp. 41-43: local coordinates at simple roots, two points at infinity for even polynomial degree, and ramification count.
Course offering: Fall 2024. Arpon Raksit is the verified instructor and personal-site host; the PDF has no named byline. The course page records a December 6 update, rather than a publication date.
Cited passages: Lecture 12, Exercise 12.1.2(1)-(2), printed page 43: box basis, comparison with the product topology, and finite-factor equality; Lecture 5, Proposition 5.3.2, printed page 21: continuity into a product is equivalent to continuity of all component functions.
Author-hosted version dated 9 October 2025. This version date is distinct from first publication.
Cited passages: §2.4, Lemma 2.4.1, pp. 23–24; §2.7, Definition 2.7.1, p. 37; §2.7, Definition 2.7.1 and linear-map construction, pp. 37–38.
Official course notes for the 2012 offering. Marco Gualtieri is identified as the course instructor; no author byline for these notes is established. University of Toronto hosts the course.
Cited passages: Topology I: smooth manifolds, Week 1, Example 1.6, equations (4)–(5), page 2; specialize the sphere charts to dimension one; Topology I: smooth manifolds, Week 1, Section 1.1, Definition 1.1, page 1.
Official course notes for the 2012 offering. Marco Gualtieri is the course instructor, rather than an inferred author of these notes.
Cited passages: Topology I: smooth manifolds, Week 2, Section 1.2, equation (11) and Definitions 1.15–1.16, pages 5–6; Topology I: smooth manifolds, Week 2, Section 1.3, equation (21), Definition 1.21 and Remark 1.24, page 7.
Author-hosted sample of Chapter 1, Introduction. The second-edition book is credited to John M. Lee, published by Springer in Graduate Texts in Mathematics 202, and carries copyright 2011. Its Chapter 1 definition is explicitly provisional; formal manifold hypotheses in this wiki are supported by the separately cited Toronto notes.
Cited passages: Introduction to Topological Manifolds by John M. Lee, second edition, Chapter 1, pages 1–3 on local models and pages 5–6 on topological invariants.
The front page is titled Convex geometry and identifies course BMETE94AM22, 2025/26 spring, updated 2 June 2026. BME’s geometry website hosts these course notes. No individual author byline is present on the inspected front page, and no instructor is assigned authorship from the URL.
Cited passages: §6, Definitions 6.2 and 6.6 and Proposition 6.7, printed pp. 36–37 (PDF page indices 35–36). No source illustration or Example 6.8 is reused; the batch’s examples, proofs and figures are original.
By Leonhard Euler. Latin journal article, written in 1750 and published in 1758 in Novi Commentarii academiae scientiarum Petropolitanae 4, 109–140; Eneström number E230. The Euler Archive at the University of the Pacific supplies the bibliographic record.
By Ravi Vakil. The title page identifies the November 18, 2017 draft. Author-hosted PDF: math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf
Consulted passages: §0.4 (goals and polynomial equations), Chapter 3 (affine schemes), and the sections on rational maps and birational morphisms.
By Edward Witten. Mathematical Research Letters 1 (1994), 769–796. Author preprint: arxiv.org/abs/hep-th/9411102
Locator: §2, equations, moduli spaces and invariants.
Locator: introduction, real Sylvester-Gallai theorem and the extensions to complex and quaternionic projective spaces.
The book page identifies the second edition as 2015; the consulted draft title page is dated 19 June 2020. These are distinct dates. Cited passages: §3.3, Theorem 3.8; §3.5, leading coefficient and volume; §3.7, Theorem 3.23; §4.3, Theorem 4.1 and its proof, Ehrhart reciprocity.
Cited passage: §18.3, Definition 18.7, printed p. 109: the compatible associative and Lie products defining a Poisson algebra. The preceding Definition 18.5 and Theorem 18.6 connect the geometric bracket to this algebraic structure.
Cited passages: section 2, Loop's algorithm and original masks; sections 3–4, especially Remark 1, original central weight alpha, lambda0=5/8 and lambda1=3/8+(1/4)cos(2pi/n); introduction, regular and extraordinary vertices.
Master's thesis, Department of Mathematics, University of Utah. The publication page confirms author, title and year; its large PDF was not accessible through the browser during this review. The original-mask formulas are instead checked against Loop's accessible 2000 report.
Cited passages: section1.2, Theorems1.7,1.10,1.12 and Example1.13; reflexivity, contact loci and nonlinear curves.
Cited passage: section3.3, Theorem3.3.4 (Ehresmann), printed pages63–64 in the current author-hosted version. No publication date is inferred.
Cited passage: section1.9, printed pages42–45, Hodge theory and harmonic curvature of line-bundle connections.
Cited passage: Theorem3.7, exceptional cobordism blowup maps; section10.1 for the mixed-invariant application. The arXiv version is distinguished from its later journal publication.
By Dennis Sullivan. Publications Mathématiques de l’IHÉS47,269–331. DOI: doi.org/10.1007/BF02684341
§§2,5 and10 develop minimal differential algebras, minimal models and their relation to rational homotopy.
Theorem1.3 gives the homology sequence and fundamental-group action; Theorem1.14, printed page24, gives cohomology, differential bidegrees and convergence to successive filtration quotients.
§1, first page, defines the Samelson product as the topological-group commutator descended from a Cartesian product to a smash product, with commutator convention f inverse, g inverse, f, g.
By Benson Farb and Dan Margalit. The inspected primary text identifies itself as Version5.0; the preface is dated January2011, rather than a publication date inferred for another edition. Institutional copy: pagine.dm.unipi.it/~a019210/Farb%20Magalit_Primer%20on%20Teichmuller%20theory.pdf
§3.1 defines supported annular Dehn twists; Proposition3.2 in §3.2, printed pages72–73, proves the exact intersection formula for every integer power. §2.3 treats the Alexander method and its filling hypotheses.
The definition distinguishes general common-fixed-point symmetry from the lattice-preserving crystallographic subclass; the occurrence section discusses crystal, vector and molecular/site interpretations.
Abstract and section 1 define uniformity by regular faces and a single vertex orbit under spatial symmetries; section 2 distinguishes the infinite prism and antiprism families.
Sections 1.2 and 5 give the thirteen exceptional convex cases, with prisms and antiprisms as separate infinite families.
Chapter 13, Definition 13.2.3 and teardrop example, p. 304; Definition 13.2.5, p. 307.
Appendix B.1, Definition B.2 and Proposition B.6, p. 66. Uses the negative of the distance-sign convention in our entry.
Annals of Mathematics 99(1), pp. 48–69. Theorem 2.12, p. 51; equation (3.1) and Proposition 3.2, p. 52.
Chapter 15, Theorem 15.3: the total Pontryagin class product formula holds modulo elements of order two. Verified against the original-text transcription in the TeXromancers edition: aareyanmanzoor.github.io/assets/books/characteristic-classes.pdf
The original scan exceeds the web extractor size limit; the theorem statement and proof were read in the retypeset original text. Locate by theorem number because the retypeset pagination differs.
Geometry and Topology Monographs 16, pages 215–282. DOI: doi.org/10.2140/gtm.2009.16.215
Sections 2 and 5 fix the countably infinite inner-product universe, finite-subspace indexing and the strict indexed-spectrum model. The text uses the mathematical symbol E infinity in its title; the source-card spelling keeps a stable readable identifier.
The opening note identifies this as an expanded version intended for a second edition, with standalone pages 1–28. Theorem A.6, pages 6–7, identifies compactly generated products with cellular products and states ordinary-product sufficient conditions. The opening construction gives the zero-skeleton/positive-cell attachment convention.
Section 2.2; Corollary 2.12, page 55. Author landing page pi.math.cornell.edu/~hatcher/VBKT/VBpage.html verifies version and unfinished status.
Section 1, pages 2–3: Seifert-matrix conditions, metabolic equivalence, invertible representatives and associated isometric structures.
Slides 12–21, especially slide 21 Dehn region/crossing figure. Author names retained as verified initials; no guessed expansion of S. Kim.
Main theorem Section 6; Corollary 2 p487; pp471–472 explain punctured/open and bounded-punctured fibers.
Definition 2.1 and Figure 2, pp396–397; Section 2 regular isotopy and writhe normalization. DOI10.1016/0040-9383(87)90009-7
Sections 4.1–4.2 pp13–14 Euler structures; Lemma5.1 and Section5.2 p18 coefficient-sensitive solid-torus filling.
Section 1 states the compact orientable setting and oriented closed-geodesic multiplicities. Section 2, equation (1), page 3, gives the Selberg trace formula with its spectral normalization; the following discussion identifies primitive oriented geodesics and iterates.
New York Journal of Mathematics 14 (2008), pages 193–204. Theorem 1 distinguishes orientation-preserving and orientation-reversing closed geodesics for nonorientable surfaces. Sections 2–3 explain their different trace weights.
Definition 2.1, page 4, uses cubical cells with face identifications. Theorem 1.1, page 2, specifies the hyperplane conditions for a local isometry to a right-angled Artin Salvetti complex.
Section 1, pages 2–3, defines finite-separation wallspaces and explicitly calls their wall-count function a pseudometric. The arXiv version is dated 4 October 2004; the PDF also displays a later compilation date, which is not used as publication metadata.
Princeton University Press book; inspected title page dates the edition 1963. Institutional scan: pagine.dm.unipi.it/~a019210/Milnor_Morse%20theory.pdf
Lemma 2.2 gives the exact smooth quadratic normal form and its completion-of-squares proof. Corollary 2.3 gives isolation of nondegenerate critical points and contrasts degenerate examples such as x cubed.
Second edition, Graduate Texts in Mathematics 218, Springer. DOI: doi.org/10.1007/978-1-4419-9982-5
Theorem 9.44 and the paragraph immediately preceding it define local flow commutation on entire time rectangles. The text explicitly distinguishes this from merely requiring both endpoint compositions to exist for incomplete vector fields.
Consulted passages: Part I §§2.1, 2.5–2.8 (arc length, curvature, torsion, spherical frames); Part VI §§7.1–7.5 (boundary terms, geodesic polygons, global Gauss-Bonnet and proof).
Consulted passages: Theorem 4.12, printed pp.458–459; holomorphic Lefschetz formula with nondegenerate fixed points.
Consulted passages: Chapter I, Cauchy formula (3.2), fundamental solution Corollary (3.4) and support caveat Remark (3.5), printed p.21; Chapter II, hypersurface Theorem (6.6), printed p.106; multiplicative meromorphic sheaf and divisor sequence (6.15), printed p.110.
Theorem 4.1, p.1: explicit Appel-Humbert datum, quotient action and Picard-group isomorphism; the printed lecture heading gives February 6 but no year.
Lecture 8, Theorem 3, printed p.135: Kähler reduction at zero; central-level qualification, p.136; scalar-circle projective-space example, pp.136–138. The cover identifies the 1991 Park City lectures and the corrected unofficial version of February 19, 2003.
Example (I.5.1), printed p.7: resolving a pencil of plane cubics by nine blowups, the elliptic fibration and its anticanonical fibre class. The author page identifies ETS Editrice Pisa, 1989.
University of Pau colloquium, February 16, 2017. Slide 16 of 23 contracts the tangent curvature and derives the Ricci form as minus i times the mixed derivative of the metric log determinant; slide 17 gives the Kähler-Einstein equation.
Section 2, manuscript pp.3–5: Hesse equations, common flex/base points, normal form and Hessians. Published in L’Enseignement Mathématique 55 (2009), pp.235–273. The wiki calculations explicitly use complex coefficients to avoid small-characteristic Hessian degeneracies.
Sections 1.1–1.3: plurigenera and birational invariance; Definition 1.2.1 and Proposition 1.2.4, Iitaka dimension and polynomial growth along sufficiently divisible powers; Definition 1.3.1 and Examples 1.3.2–1.3.7, Kodaira dimension and curve/surface examples.
The author and course are verified by the institutional page people.math.harvard.edu/~mpopa/271/index.html; no publication year inferred.
Appendix A, pages 119–121 in the displayed text (PDF pages 119–121): fractional densities, their product and integration, and the intrinsic Hilbert space obtained by completing compactly supported half-densities. The preface identifies courses in 1992 and 1994; no publication year inferred from those course dates.
Publications Mathématiques de l’IHÉS 5, pages 5–64. DOI: doi.org/10.1007/BF02684746
Final section, pages 63–64, especially Satz 5: local freeness and bijective fiber-cohomology comparison under constant fiber dimension; Satz 6 records Euler-characteristic constancy under flatness and a reduced connected base. Indexed original passages were read; direct PDF fetching timed out. Archive metadata identifies a later correction, which should be retained alongside the original bibliographic record.
Based on CIME lectures in Cetraro, September 2003. Sections 1–2: symplectic background, Lefschetz pencils and fibrations, local models, blowup of base points, and algebraic generic pencils. The date on the consulted arXiv version is 4 January 2004; this is distinct from the later published chapter.
The symplectic-sum construction glues along codimension-two symplectic submanifolds with identified opposite normal bundles. The publisher abstract verifies the resulting realization of finitely presented groups in every even dimension at least four. For the local gluing model and normal-bundle conditions, the Auroux–Smith lectures give a complementary primary exposition.
Annals of Mathematics 120(1), 39–87. Theorem A, printed page 39, proves weak contractibility of the pointed mapping space from the classifying space of a locally finite discrete group to a connected finite-dimensional CW complex.
Lecture 22, 27 April 2010, from the author-hosted Chromatic Homotopy Theory course. Definition 5 and the following calculation on page 3 define Morava K-theory and its coefficient ring; the course index www.math.ias.edu/~lurie/252x.html identifies the author-hosted series.
Author lecture text, based on a Hamburg 2003 course. Sections 11 and 13 give the Ricci form and Kähler–Einstein equation; sections 7 and 17 discuss projective and Ricci-flat examples.
Primary lecture text: arxiv.org/pdf/1008.0973
Section1 defines the tangent-plus-cotangent bundle, its natural pairing and Courant bracket; section3 explains ordinary complex and symplectic structures as generalized complex structures.
Course lecture text. Definitions1.3–1.4 specify the k-space and weak Hausdorff conventions; section2, Lemmas2.1–2.2, defines the kified product and its universal property.
Edited in modern notation by Thomas Little Heath, Cambridge University Press, 1896. Primary translated/adapted Conics text; the edition rearranges the ancient work. Cited passage: Book I, conic sections and their defining geometric relations.
Primary work in French translation. Propositions I and III give the circle-area relation and the rational circumference-to-diameter bounds 223/71 and 22/7. Indexed translated text inspected; direct fetching returned 403.
Annals of Mathematics 48(2), 290–320. Primary publisher record: www.jstor.org/stable/1969172 . Original paper introducing cochain constructions underlying the mod-two cohomology operations. Full publisher text was inaccessible; bibliographic record and later primary mathematical expositions identify the contribution.
Verified locator: Lecture 20, Definition 2, Example 4 and Example 7; universal property and rational localization.
By J. M. Landsberg.
Verified locator: Section 6: joins and secant varieties. The byline is J. M. Landsberg.
Verified locator: 1998 paper, abstract and introduction: Stein surfaces and contact three-manifolds.
Verified locator: Abstract and introduction: proof sketch of the cobordism hypothesis and classification of extended field theories.
Verified locator: Inventiones Mathematicae 82 (1985), 307–347; section 0.3.A, Corollary, printed page 310: symplectic ball embedding obstruction.
Verified locator: Pacific Journal of Mathematics 195(1), September 2000, pages 179–205. Introduction and Theorem 1: equations, perturbations, wall crossing and b1=0 correction term.
Annals of Mathematics 161, 1093–1110. Introduction and Theorem 1.1: boundary distances determine a simple compact surface metric up to a boundary-fixing diffeomorphism.
Verified locator: Annals of Mathematics 86(2), September 1967, pp.374–407; introduction, isolated fixed-point formula.
Verified locator: Author prepublication manuscript, front matter copyright 2016; sections 1.1, 2.1, 4.7–4.8: bilinear complexity, tensor rank and border rank.
Verified locator: Chapter 4, section 4.1: incidence structures, typed elements, relations and distinction from incidence geometries. The article uses the point-block, two-type convention.
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