# Tag Info

19

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 Poincaré's Analysis situs and related papers he explains the motivation in one of the introductory paragraphs, written by himself: Poincaré set the agenda ...

14

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 about three houses and three wells belongs to this class. One notable invention was Möbius band, another Gauss knot invariant. However, one can agree with ...

11

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 in the early 1870s. According to an account by Zöllner (1878a), Klein spoke with him during a scientific conference on this subject shortly before publishing a ...

10

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 called "qualitative methods" which are based to a large extend on topology. For example, the existence of periodic orbits. In the introduction to his paper Analysis ...

9

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 $Y$ depends only on (what we now call cohomology of) its fundamental group $\pi=\pi_1(Y)$: [Since] homology and cohomology groups of $Y$ (with coefficients in $A$) ...

8

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 complex functions, e.g. $f(z) =z^{1/2}$ are multi-valued in certain subdomains of the complex plane, so when trying to continue along the closed curve one might ...

8

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 solution). Euler first used the concept in the first half of the 18th century: Perhaps the first work which deserves to be considered as the beginnings of ...

8

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 special class of groups has long been known and used. A detailed computation of the invariant integral on $\mathfrak{SD}(n)$ was given in 1897 by HURWITZ [1]. ...

7

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 methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. The purpose of this book is to ...

7

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. I will give you a translation here, and some comments of mine. Beforehand, though, I think it is fair to say hat you seem to have been misled when you wrote ...

7

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. Hausdorff started with metric spaces, but then generalized them. Of course, the background to Hausdorff's work was the 19th work on continuity, and the so-called "...

7

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, Proceedings of the section of mathematical sciences, 14.

6

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 Situs and Algebraic Geometry, but his influential 1930 monograph is already Topology, which he defines as "doctrine of the modal features of objects, or of the ...

6

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 theorem, some "algebraic" and some "geometric". The first geometric proof was given by Baer and Levi in 1936, and this is the proof we present. There is another ...

6

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. 353–365. And two years later he published a proof that a function continuous on a closed interval is uniformly continuous there in Die Elemente der Functionenlehre ...

6

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 paper “Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten,” Rendiconti del Circolo Matematico di Palermo, 25 (1908), Erhard Schmidt defines ...

5

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-dire homéomorphe à l'hypersphère. (Stillwell, p. 179. Same on pp. 141, 152, 169, 255. Stillwell apparently missed the subtlety you note that what is said on p....

5

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). Kline (1972, p. 950) says Bolzano’s motivation was to rigorously prove the fundamental theorem of algebra, and on his way, the intermediate value theorem — ...

5

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 gives his definition of the fundamental group. To this end he starts by almost copying his 1883 article which we have just discussed. But now he introduces a group! ...

4

I don't know exactly who introduced the terminology, but "separable" comes from functional analysis, probably Hilbert spaces, where "having a countable dense subset" and "having a countable basis of open sets" are equivalent (as for metric spaces in general). I agree that now most, if not all, authors seem to use "separable" for topological spaces with the ...

4

Hero mentions the work of Dionysodorus in his "Metrica" this way: "διονυσωδώρω έν τω περί τησ σπείρασ έπιγραφομένω", namely "Dionysodorus in his work On σπείρασ". In modern Greek σπείρασ stays for "spirals", but was used in ancient Greek for the Latin torus. You can find the original text here at page 128, line 3 and 4.

4

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 19th century. In Cauchy and Puiseux: Two Precursors of Riemann Papadopoulos even mentions earlier authors, especially Puiseux: "Riemann defined these surfaces ...

4

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 http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Topology_in_mathematics.html will be of interest for aspects of topology before Poincaré came on the scene.

4

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 out of print. Used copies go for about a hundred quid. I suppose Dover might look kindly on a request for a reprint. This is very probably also relevant (and it ...

4

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 for subsets of real line, plane, and then curve and function spaces (Cantor's accumulation points, derived and closed sets on the line, Weierstrass's theorems, ...

4

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, volume 3, edited by C.E. All and R. Lowen

3

You wrote "topology was not developed to [solve?] problems but to gain insight to foundations of mathematics". This is completely wrong, and the notion of covering is a good example. Klein describes in his memoirs how he discovered what is now called the Uniformization theorem. (This is a very concrete statement from the theory of analytic functions). Then ...

3

There is a famous paper by Alexandroff & Urysohn, I believe, where they analyze many compactness-type definitions for the recently-defined notion of "topological space". Since "compact" was already used to mean: every sequence has a convergent subsequence, they came up with other names. They used the term "bicompact" for: every open cover has a finite ...

3

The earliest occurence Milnor mentions in his survey Differential Topology Forty-six Years Later is Whitehead's paper On the Homotopy Groups of Spheres and Rotation Groups (1942). If $J_n$ is the image of the stable Whitehead homomorphism from the $n$th stable homotopy group of rotation groups to the $n$th stable homotopy group of spheres, then $J_n$ is ...

3

Topological spaces appear to have their roots in the nineteenth century. It started, indirectly, with the theory of limits and delta-epsilon proofs. A major breakthrough occurred with the development of set theory (e.g. DeMorgan's Laws) in the middle to latter part of the century. This led to the "generalization" of limit, convergence, and accumulation point ...

Only top voted, non community-wiki answers of a minimum length are eligible