"Not just an idle game":(examining some historical conceptual arguments in homotopy theory)
““Not just an idle game”(examining some historical conceptualarguments in homotopy theory)
Ronald BrownOctober 29, 2020
Abstract
Part of the title of this article is taken from writings of Einstein. which argues that we need to exerciseour ability to analyse familiar concepts in science, and to demonstrate the conditions on which their justi-fication and usefulness depend, and the way in which these developed, little by little . . . . The aim is to dothis for the concepts of (i) the fundamental group of a pointed space, due to Poincar´e; (ii) the fundamentalgroupoid of a space; (iii) the fundamental groupoid of a space with a set of base points, introduced by theauthor in 1967; and (iv) the search for higher dimensional versions of the fundamental group.The history goes back at least to the ICM meeting in Z¨urich in 1932; the initial negative reactions to theseminar by E. ˇCech on higher homotopy groups; then the subsequent work of Hurewicz on these groups;the influence of this and subsequent work on the notion of space in topology; and also generalisations ofthe theorem of Van Kampen.
Introduction
Part of the title of this article is taken from writings of Einstein (1879-1955) in the correspondence publishedin [22]: . . . the following questions must burningly interest me as a disciple of science: What goal willbe reached by the science to which I am dedicating myself? What is essential and what is basedonly on the accidents of development? . . . Concepts which have proved useful for ordering thingseasily assume so great an authority over us, that we forget their terrestrial origin and accept themas unalterable facts. . . .
It is therefore not just an idle game to exercise our ability to analysefamiliar concepts, and to demonstrate the conditions on which their justification and usefulnessdepend, and the way in which these developed, little by little . . .
This quotation is about science, rather than mathematics, and it is well known for example in physics thatthere are still fundamental questions, such as the nature of dark matter, to answer. There should be awarenessin mathematics that there are still some basic questions which have failed to be pursued for decades; thus weneed to think also of educational methods of encouraging their pursuit. More information on most of the people mentioned in this article may be found in the web site https://mathshistory.st-andrews.ac.uk/Biographies/ . a r X i v : . [ m a t h . A T ] O c t Homotopy groups at the ICM Z ¨urich, 1932
Why am I considering this ancient meeting? Surely we have advanced since then? And the basic ideas havesurely long been totally sorted?Many mathematicians, especially Alexander Grothendieck (1928-2014), have shown us that basic ideascan be looked at again and in some cases renewed.The main theme with which I am concerned in this paper is little discussed today, but is stated in [30]: itinvolves the introduction by the well respected topologist E. ˇCech (1893-1960) of homotopy groups π n ( X, x ) of a pointed space ( X, x ) , and which were proved by him to be abelian for n > . Because of this abelianproperty ˇCech was persuaded by Heinz Hopf (1894-1971) to withdraw his paper, so that only a small para-graph appeared in the Proceeding [18]: it was argued that these groups were inappropriate for what was a keytheme at the time, the development of higher dimensional versions of the fundamental group π ( X, x ) of apointed space as defined by H. Poincar´e (1854-1912). In many of the known applications of the fundamentalgroup in complex analysis and differential equations, the largely nonabelian nature of the fundamental groupwas a key factor. However. the abelian homology groups H n ( X ) were known to be well defined for any space X , and that if X was path connected, then H ( X ) was isomorphic to the fundamental group π ( X, x ) madeabelian. Indeed P. Alexandrov (1896-1998) was reported to have exclaimed: ”But my dear ˇCech, how canthey be anything but the homology groups?” Remark 1.1
It is useful to give for n = 2 and in a modern form the argument for the abelian nature of thehomotopy groups π ( X, x ) . Proof
We write I for the unit square [0 . , and define G = π ( X, x ) as the homotopy classes rel theboundary of I of maps I → X which take all of the boundary ∂I of I to the base point x . It is then easyto define, analogously to the fundamental group, two compositions in “directions” and which we write as a ◦ b, a ◦ b . Both compositions have the structure of a group.. We also have an axiom, called the “interchangelaw”, that each of these compositions is a morphism for the other: that is, for any a, b, c, d ∈ G ( a ◦ b ) ◦ ( c ◦ d ) = ( a ◦ c ) ◦ ( b ◦ d ) (1)which can be interpreted in matrix notation by writing (cid:20) ab (cid:21) = a ◦ b, (cid:2) a c (cid:3) = a ◦ c as that there is only one interpretation of the matrix of compositions (cid:20) a cb d (cid:21) . (cid:15) (cid:15) (cid:47) (cid:47) In this way a “2-dimensional formula” is easier to interpret than the “1-dimensional formula” (1).Now let e , e be the identities for ◦ , ◦ . Considering the matrix of compositions (cid:20) e e e e (cid:21) I heard of this comment in Tbilisi in 1987 from G. Chogoshvili (1914-1998) whose doctoral supervisor was Alexandrov. Itshould also be said that Alexndrov and Hopf were two of the most respected topologists: their standing is shown by their invitationby S. Lefschetz (1884-1972) to spend the academic year 1926 in Princeton. e = e . So we write e for either. Now interpreting in turn the matrices (cid:20) a ee b (cid:21) (cid:20) e ab e (cid:21) gives first that a ◦ b = a ◦ b , so each are written ab , and second that ab = ba . This can be interpreted as“higher dimensional groups are abelian groups”. Remark 1.2
Note however that this argument fails if some composition e ◦ e is not defined. This is relevantto Section 3.W. Hurewicz (1906-1956) was at this ICM. With the publication of two notes [29] which shed light onthe relation of the homotopy groups to homology groups, the interest in these homotopy groups started;with the growing study of the complications of the homotopy groups of spheres, which became seen as amajor problem in algebraic topology, the idea of generalisations of the nonabelian fundamental group becamedisregarded, and it became easier to think of “space” and the “space with base point” necessary to define thehomotopy groups, as in substance synonymous.In 1968, Eldon Dyer (1934-1997), a topologist at CUNY, told me that Hopf told him in 1966 that thehistory of homotopy theory showed the danger of people being regarded as “the kings” of a subject and sokey in deciding directions. There is a lot in this point, cf [1].However it seems to be true that Aleksandrov and Hopf were correct in suggesting that the abelian ho-motopy groups are not what one would really like for a higher dimensional generalisation of the fundamentalgroup! That does not mean that the higher homotopy groups are without interest; nor does it mean that thesearch for such higher dimensional generalisations should be completely abandoned. One reason for this interest in fundamental groups was their known use in important questions relating com-plex analysis, covering spaces, integration and group theory. H.Seifert (1907-1996) proved useful relationsbetween simplicial complexes and fundamental groups, [36], and a paper by E. H. Van Kampen (1908-1942)[31] gave a general result applied to the complement in a 3-manifold of an algebraic curve. However a modernproof was given by R.H. Crowell (1928-2006) in [20] following lectures of R.H. Fox (1913-1973). The resultis often called the Van Kampen Theorem (VKT) and there are many excellent examples of applications of itin expositions of algebraic topology.The usual statement of the VKT for the fundamental group is as follows.
Theorem 2.1
Let the space X be the union of open sets U, V with intersection W ad assume U, V, W arepath connected. Let x ∈ W . Then the following diagram of fundamental groups and morphisms induced byinclusions: π ( W, x ) i (cid:15) (cid:15) j (cid:47) (cid:47) π ( V, x ) k (cid:15) (cid:15) π ( U, x ) h (cid:47) (cid:47) π ( X, x ) , (2) is a pushout diagram of groups. G is a group and f : π ( U, x ) → G, g : π ( V, x ) → G are morphisms of groups such that f i = gj , then thereis a unique morphism of groups φ : π ( X, x ) → G such that φh = f, φg = k . This property is called the“universal property” of a pushout, and proving it is called “verifying the universal property”. In particular,such a verification does not involve a particular construction of the pushout, nor a proof that all pushouts ofmorphisms of groups exist.The limitation to path connected spaces and intersections in Theorem 2.1 is also very restrictive. . Becauseof the connectivity condition on W , this standard version of the Van Kampen Theorem for the fundamentalgroup of a pointed space did not compute the fundamental group of the circle, which is after all the basicexample in topology; the standard treatments instead make a detour into a small part of covering space theoryby introducing the winding number of the map p : R → S , t (cid:55)→ e πit from the reals to the circle. This is a theme with which I became involved in the years since 1965.A groupoid is defined in modern terms as a small category in which every morphism is an isomorphism.It can be considered as a “group with many identities”, or more formally as an algebraic structure with partialalgebraic operations, [26]. I like to define “higher dimensional algebra” as the study of partial algebraicstructures where the domains of the algebraic operations are defined by geometric conditions.The simplest non trivial example of a groupoid is the groupoid say I which has two objects , andonly one nontrivial arrow ι : 0 → . and hence also ι − : 1 → . This groupoid looks “trivial”, but it isin fact the basic “transition operator”. Groupoids had been defined by Brandt (1886-1954) in 1926, [2], inextending work of Gauss (1777-1815) on compositions of binary quadratic forms to the quaternionic case;their use in topology had been initiated by K. Reidemeister (1893-1971) in his 1932 book, [35]. The use ofthe fundamental groupoid π ( X ) of a space X , defined in terms of homotopy classes rel end points of paths x → y in X was a commonplace by the 1960s. Students find it quite easy to see the idea of a path as a journey,which may be from a to b rather then just from a to a .I was led to Philip Higgins’ (1924-2015) paper on groupoids, [27]. I noticed that he utilised pushouts ofgroupoids, and so decided to insert in the book I was writing in 1965 an exercise on the Van Kampen Theoremfor the fundamental groupoid π ( X ) . Then I thought I had better write out a proof, and when I had done so itseemed so much better than my previous attempts that I decided to switch the exposition to groupoids ratherthan groups.But I was still annoyed that I could not deduce the fundamental group of the circle. I then realised wewere in a “Goldilocks situation”: one base point was too small; taking the whole space was too large; but forthe circle two base points were just right! So, we needed a definition of the fundamental groupoid π ( X, S ) for a set S of base points chosen according to the geometry of the situation. This was published as the paper[3] and is in all editions of [4], as well as in [28].But my own impression is that once it had become sure that homotopy groups were very important, thefact that they depended on choosing a single base point, made it more difficult to accept a wider choice.A conversation with G.W. Mackey (1916-2006) in 1957 at a Swansea BMC informed me of the notion of“virtual groups”, cf [33, 34], and then led me to extensive work of C. Ehresmann (1905-1979), and his school, Comments by Grothendieck on this restriction are quoted extensively in [5], see also [25, Section 2], and there is a generaldiscussion of many base points on Mathoverflow at https://mathoverflow.net/questions/40945
As we have shown, “higher dimensional groups” are just abelian groups.However this is no longer so for “higher dimensional groupoids”.It seemed to me in 1965 that some of the arguments for the VKT generalised to higher dimensions andthis was prematurely claimed as a theorem in [3]. One of these arguments comes under the theme or slogan of “algebraic inverses to subdivision”. ↔ From left to right gives subdivision . From right to left should give composition . What we need for higherdimensional, nonabelian, local-to-global problems is:
Algebraic Inverses to Subdivision.
This aspect is clearly more easily treated by cubical methods than the standard simplicial or globular ones.One part of the proof of the VKT for the fundamental group, namely the uniqueness of a universal mor-phism, is more easily expressed in terms of the double groupoid (cid:3) G of commutative squares in a group orgroupoid G , which I first saw defined in [21]. The essence of its use is as follows: consider a diagram ofmorphisms in a groupoid: b a (3)Suppose each individual square is commutative, and the two vertical outside edges are identities. Then weeasily deduce that a = b .For the next dimension we therefore expect to need to know what is a “commutative cube”, and this isexpected to be in a double groupoid in which horizontal and vertical edges come from the same groupoid: G = G . (cid:5) (cid:47) (cid:47) (cid:15) (cid:15) (cid:5) (cid:15) (cid:15) (cid:5) (cid:63) (cid:63) (cid:47) (cid:47) (cid:15) (cid:15) (cid:5) (cid:63) (cid:63) (cid:15) (cid:15) (cid:5) (cid:47) (cid:47) (cid:5)(cid:5) (cid:47) (cid:47) (cid:63) (cid:63) (cid:5) (cid:63) (cid:63) (4)We want the “composed faces” to commute! What can this mean? It could be more correctly called “ ideas for a proof in search of a theorem”.
5e might say the top face is the composite of the other faces: so fold them flat to give the left handdiagram of Fig. 1, where the dotted lines show adjacent edges of a “cut”. We indicate how to glue these edgesback together in the right hand diagram of this Figure by means of extra squares which are a new kind of“degeneracy”. a ba b − c − c d d a ba b − c − c d d Figure 1: “Composing” five faces of a cubeThus if we write the standard double groupoid identities in dimension 2 aswhere a solid line indicates a constant edge, then the new types of square with commutative boundaries arewritten . These new kinds of “degeneracies” were called connections in [16], because of a relation to path connectionsin differential geometry. In a formal sense, and in all dimensions, they are constructed from the two functions max , min : { , } → { , } .It is explained in Section 8 of [8] how the introduction of these “connections” to the traditional theoryof cubical sets remedied some main perceived (since 1955) deficiences of the cubical as against the standardsimplicial theory; yet the change allowed better control of homotopies and higher homotopies (because of therule I m × I n ∼ = I m + n ). It crucially allowed “algebraic inverses to subdivision”, and so possibilities for HigherVan Kampen Theorems, starting in dimensions 1, 2 in [11, Part 1], which amounts to a rewrite of traditionalsingular and cellular algebraic topology.We explain briefly the basic definition of π ( X, A, a ) . the second relative homotopy group of a basedpair, defined as the homotopy classes relative to a base point say of I , of maps I → X which take { } × I ∪ { , } × I to a . The composition of such classes is in the direction : aa X X X X a X aA A A A A (5)Whitehead proved that the boundary ∂ : π ( X, A, a ) → π ( A, a ) and an operation of the group π ( A, a ) on π ( X, A, a ) have the structure of what he called a crossed module ; these structures have proved importantin studying higher groupoids. He proved in [37, Section 16] what we call Whitehead’s free crossed moduletheorem that in the case X is formed from A by attaching 2-cells, then this crossed module is free on thecharacteristic maps of the 2-cells.This definition of this crossed module involves several choices: which vertex to choose as the base pointof the square; which edges of the square should map to the base point a , so that the remaining edge mapsinto A . However it is a good principle to reduce, preferably completely, the number of choices used in basicdefinitions (though such choices are likely in developing consequences of the definitions). The paper [10],6ubmitted in 1975, contained the definition of a structure for a triple of spaces X ∗ = ( X , X , X ) of spaces;this (cid:37) ( X ∗ ) consisted in dimension of X (as a set); in dimension of π ( X , X ) ; and in dimension of homotopy classes relative to the vertices of maps ( I , sk I , sk I ) → ( X , X , X ) , where sk r K of aCW-complex K denotes the space of cells of dimension (cid:54) r . Note that this definition makes no choice of preferred direction. It is fairly easy and direct to prove that (cid:37) ( X ∗ ) may be given the structure of double groupoid with connection containing the crossed module over agroupoid π ( X ∗ ) . That is, the proofs of the required properties of (cid:37) to make it a -dimensional version of thefundamental group as sought in the 1930s are fairly easy but not entirely trivial. The proof of the correspond-ing Van Kampen Theorem allows a nonabelian result in dimension 2 which substantially generalises the workof Whitehead on free crossed modules.An account of an analogous theory in all dimensions, based on filtered spaces, is given in [11], with furtherbackground and motivation in [8].To get nearer to a fully nonabelian theory we so far have only the use of n -cubes of pointed spaces as in[13, 14, 6]. It is this restriction to pointed spaces that is a kind of anomaly, and has been strongly criticisedby Grothendieck as not suitable for modelling in algebraic geometry. However, the paper [23] gives anapplication to a well known problem in homotopy theory, the first non-vanishing homotopy group of an n -ad ;also the nonabelian tensor poduct of groups from [13] has become a flourishing topic in group theory (andanalogously for Lie algebras); a bibliography . Note that the mathematical notion of group is deemedfundamental to the idea of symmetry, whose implications range far and wide. The bijections of a set S forma group Aut ( S ) . The automorphisms Aut ( G ) of a group G form part of a crossed module χ : G → Aut ( G ) .The automorphisms of a crossed module form part of a “crossed square” [9]. These structures of set, group,crossed module, crossed square, are related to homotopy n -types for n = 0 , , , .The use in texts in English on algebraic topology of sets of base points for fundamental groupoids seemscurrently restricted to [4], which itself has been termed “idiosyncratic”. The argument over ˇCech’s seminarto the 1932 ICM seems now able to be resolved through this development of groupoid and higher groupoidwork, and he surely deserves credit for the first presentation on higher homotopy groups, as reported in[18] . References [1] Aleksandrov, P. S. and Finikov, S. P. ‘Eduard ˇCech: Obituary.’
Uspehi Mat. Nauk (1 (97)) (1961)119–126. (1 plate). 3[2] Brandt, H. ‘ ¨Uber eine Verallgemeinerung des Gruppenbegriffes’. Math. Ann. (4) (1926) 360–366. 4[3] Brown, R., “Groupoids and Van Kampen’s theorem”, Proc. London Math. Soc. (3) (1967) 385-401.4, 5 In that paper it was assumed that each loop in X is contractible in X but this later proved inconvenient and so the use ofhomotopies fixed on the vertices of I and in general I n was used in [11]. See This assertion is supported by a web search on “Institute of higher structures in maths”. cf. Ecclelsiastes 9.11.
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