19 votes

How did Poincaré discover the fundamental group?

The concept of fundamental group was introduced in Poincaré's seminal paper Analysis situs, so that is a natural place to look for some motivation for this concept. In Stillwell's nice translation of ...
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  • 3,681
14 votes

Who created topology, and when, and what problems led to this creation?

Some topological problems were considered long ago, for example by Euler (see the previous answer). Some ideas about topology were even earlier proposed by Leibniz. The famous kindergarten problem ...
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11 votes
Accepted

A knot cannot be tied in 4-dimensions, but when was this conjectured and proven?

A nice account is found in a note to R. Steiner's Die vierte Dimension (1995; translation): Felix Klein (1845–1925) seems to have been the first mathematician to draw attention to this phenomenon ...
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10 votes
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What specific problems motivated Poincaré's work on topology?

One motivation, perhaps the principal one, was his work in ordinary differential equations. (And celestial mechanics, as an application of ordinary differential equations). He introduced what is ...
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10 votes
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(Co)Homology: From topology to the rest of mathematics?

I'd recommend Weibel’s History of homological algebra (1999)(pdf). He describes many threads, such as roots of group cohomology in Hurewicz’s observation that cohomology of an aspherical space $Y$ ...
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9 votes

Origins and history of branched covering

The theory of branched (or ramified) coverings has its origins in continuation of analytic functions and the attempts to find maximal analytic continuations of a given function. However, certain ...
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8 votes
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Who created topology, and when, and what problems led to this creation?

Many sources, including this one, credit the idea of topology (and its applications) to Leonhard Euler, to solve the puzzle of the Seven Bridges of Königsberg (or, rather to prove that there was no ...
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  • 8,082
8 votes

What examples led to the modern definition of a topological space?

I believe that our modern definition of a topological space came primarily from Hausdorff's book Grundzüge der Mengenlehre (Foundations of Set Theory), first published in 1914, 2nd ed. 1927. ...
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8 votes
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On the history of Haar measure

Cohn himself recommends historical notes at the end of sections 15, 16 of Abstract Harmonic Analysis by Hewitt and Ross, volume 1. Here is an excerpt: "Invariant integration on one or another ...
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8 votes

What is the source of Hermann Weyl's remark about "near-sighted policemen" with respect to compact spaces?

The closest match I could find is in H. Weyl, "Harmonics on homogeneous manifolds." Annals of Mathematics, Second Series, Vol. 35, No. 3, July 1934, pp. 486-499, as reproduced in K. ...
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8 votes
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Who proved Banach fixed point theorem in abstract metric spaces for the first time?

Pages 97-107 of the book Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles is an article by Guerraggio about Caccioppoli. On p. 100, Guerraggio says the contraction ...
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  • 4,064
7 votes
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What topological ideas did Gauss introduce to his student Möbius?

While you already accepted an answer, it seems not superfluous to add another one, in particular since you are implicitly asking for a better translation/understanding of the passage you quoted. ...
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7 votes

(Co)Homology: From topology to the rest of mathematics?

As a complement to the answer provided by Francois Ziegler, I would add the first three paragraphs of Homological Algebra (1956), by Henri Cartan and Samuel Eilenberg: During the last decade the ...
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7 votes
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Origin of Compactness

See Wiki's entry on Compact space : Alexandrov, Pavel and Urysohn, Pavel (1929), "Mémoire sur les espaces topologiques compacts", Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, ...
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7 votes

Which book covers topology historically?

J. Dieudonne, A History of Algebraic and Differential Topology, 1900 - 1960.
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7 votes

Which book covers topology historically?

I don't entirely understand the question, but this book is probably relevant: J. H. Manheim, The Genesis of Point Set Topology (1964). I must have borrowed it from a library a long time ago. It is ...
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6 votes
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What caused the name change from "analysis situs" to "topology"?

According to MacTutor, "the subject was known as analysis situs for many years and only in the late 1920s was the English word topology used by Lefschetz". Lefschetz's 1924 work is titled Analysis ...
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6 votes
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Who discovered the singular cup product?

The history of the cup product is described on pages 135–136 of Never a Dull Moment: Hassler Whitney, Mathematics Pioneer by Keith Kendig. A key event was the 1934 International Conference in Topology,...
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6 votes
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Who discovered the topological proof of Nielsen-Schreier theorem?

See the end of the Wikipedia link in your first sentence. The source is Rotman's Introduction to the Theory of Groups (1995), which reads on p.383:"There are today several different proofs of this ...
6 votes
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Who first proved the "Cantor-Heine theorem" on uniform continuity?

An explicit definition of uniform continuity was first published by Heine in Über Trigonometrische Reihen (On Trigonometric Series), Journal für die Reine und Angewandte Mathematik, 71 (1870), pp. 353–...
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6 votes

Which book covers topology historically?

I. M. James, History of Topology. This is a collection of 40 essays by different authors, on topics related mostly to manifolds and algebraic topology. Perhaps the page https://mathshistory.st-andrews....
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  • 4,064
6 votes
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Normed vector space : when and who?

A good source for these types of questions is Miller's site Earliest Known Uses of Some of the Words of Mathematics. On the norms in vector spaces we find the following: "On page 57 of his 1908 ...
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  • 66k
6 votes
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Historical ways of *presenting* or axiomatizing the notion of a topological space

An early definition of topology is given in the book Kuratowski, Topology I (first edition, 1933). It is defined in terms of a "closure operator", $X\mapsto\overline{X}$ acting on the set of ...
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5 votes
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What exactly did Poincaré mean by 'simply connected'?

In the setting of the conjecture (closed manifolds) he certainly meant “homeomorphic to the $n$-sphere” — see second page of that fifth complement: simplement connexe au sens propre du mot, c'est-à-...
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5 votes

Motivation of Continuous Functions

Isolation of the modern concept is generally attributed to Bolzano (1817; translation) and/or Cauchy (1821), with some controversy on their independence: see Grattan-Guinness (1970), Freudenthal (1971)...
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5 votes
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Who first described the fundamental group as the group of deck transformations?

The idea of a relation between fundamental groups and permutations of the universal cover long predates Grothendieck and SGA. It appears implicitly already in Riemann's work on complex surfaces in ...
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  • 66k
5 votes

Who first described the fundamental group as the group of deck transformations?

The answer to the title question is Poincaré, in the very note Sur l’Analysis situs (1892) where he first introduced the fundamental group. Cf. the description by “Saint-Gervais”: Now Poincaré ...
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5 votes

History of various definitions of topology

Here is a partial answer. By the time these definitions were introduced as definitions there was a body of previous work, where they were convenient side notions for stating theorems in special cases ...
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  • 66k
5 votes

On the history of Haar measure

Try these references: Section 7.5 of History of Topology, edited by I. M. James. Section 2.2 of the chapter "Topological Features of Topological Groups" in Handbook of the History of General Topology,...
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  • 321
5 votes

Who is the John Thomas of "Thomas's Plank" and "Thomas's Corkscrew"?

I can't speak to whether my father (John David Thomas II) is the author of Thomas' Plank and Thomas'Corkscrew, but he DEFINITELY is the John Thomas you reference that taught at NMSU. I am grateful to ...
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