A (Brief) History of Homotopy Theory

Transcription

1 April 26, 2013

2 Motivation Why I m giving this talk: Dealing with ideas in the form they were first discovered often shines a light on the primal motivation for them (...) Why did anyone dream up the notion of homotopy, and homotopy groups? Why you might be interested in listening: Mazur Homotopy as a tool preceds homotopy as a concept Homotopy groups were very elusive Ushers in transition from analysis to topology

3 What is Homotopy? Red and Blue curves: the images of a maps f, g : [0, 1] = Y X A continuous deformation from one path to the other f and g are homotopic (as maps from Y into X) if there exists a family of cts maps {h t } for t [0, 1] such that h 0 = f, h 1 = g x [0, 1], t h t (x) continuous

4 What is Homotopy? f, g : X Y, are homotopic H : X [0, 1] Y continuous s.t. H(x, 0) = f(x), H(x, 1) = g(x) Homotopy is an equivalence relation. + = t = 0 t = 1 t = 0 t = 1 0 1/2 1 It is very important to remember this is dependent on the topological spaces X Y

5 Homotopy Groups Now consider based homotopy classes of maps of the sphere into the space X: π 1 (X, x 0 ) = [S 1, ; X, x 0 ] Why does this form a group: e : S 1 x 0 : the constant map is an identity f 1 : indicates transversing f in the opposite direction [f][g] = [f g] : indicates transversing first f and then g can check associativity all of these respect homotopy

6 Origins of the concept of homotopy and the fundamental group Analysis Cauchy and mobility of path, 1825 Riemann and connectivity, 1851 Jordan and deformations of curves, Analysis Situs Poincaré and a definition of π 1 of a manifold, 1892

7 Higher homotopy groups Hopf, and a nontrivial map S 3 S 2, 1931 Cech, introduction of abstract homotopy groups, 1932 Hurewicz, higher homotopy groups and homotopy equivalence, 1935 Eilenberg and obstruction theory, 1940

8 A summary Cauchy, 1825 Riemann, 1851 Jordan, Poincaré, 1892 Hopf, 1931 Cech, 1932 Hurewicz, 1935 Eilenberg, 1940

9 A summary Cauchy, 1825 Riemann, 1851 Jordan, Poincaré, 1892 What s going on? Hmmm... Hopf, 1931 Cech, 1932 Hurewicz, 1935 Eilenberg, 1940 Oh.

10 The First Uses of Homotopy Want to integrate some complex function f between these endpoints, does the path matter? Gauss and Poisson both note 1815 that it can!

11 Cauchy s Work on Integration I = X+iY x 0 +iy 0 f(z)dz Identify complex numbers with points in the plane Introduce a mobile curve joining x 0 + iy 0 and X + iy x = φ(t), y = χ(t), monotone functions of t, continuously differentiable I = A + ib = T t 0 f(φ(t) + iχ(t)) [ φ (t) + iχ (t) ] dt Theorem The result is independent of the choice of φ and χ if f is complex differentiable for x 0 x X, y 0 y Y

12 Cauchy s Work on Integration If one wants to pass from one curve to another, which is not infinitely near the first, one can imagine a third mobile curve, which is variable in its shape, and have it coincide successively and at different instances with both fixed curves.

13 Riemann and Connectivity Definition 1 A surface is simply-connected if every cross-cut (interior arc joining boundaries) on the surface divides the surface. 2 A surface has connectivity number n if n 1 cross-cuts turn it into a simply-connected surface. surface connectivity number sphere 1 torus 3 g-hole torus 2g + 1 Note: this is not in general equivalent to saying a space is n-connected using π n = 0!

14 Riemann and Connectivity Simply Connected It does coincide with our standard definition of simply connected (π 1 (X) = 0) By classification of surfaces, we only have the plane and the sphere that are simply connected Jordan curve theorem The quantifier is very necessary, there are even not null-homotopic crosscuts that divide a surface which is not simply connected

15 Jordan and deformations of curves In 1966, published a paper on closed curves on surfaces: Definition Any two closed contours, drawn on a given surface, are called reducible into one another, if one can pass from one to the other by a progressive deformation. Any two contours drawn in the plane are reducible to one another; however this is not true on any surface: for instance, on a torus a meridian and a parallel are two irreducible contours.

16 Elementary Contours S C 0 C 0 a 0 P Γ 0 Γ 1 b A a 1 C 1 C 1

17 Elementary Contours Figure: Punctured 2-hole torus n : the maximal number of non-intersecting closed curves that do not divide the surface (what we call the genus)

18 Elementary Contours S C 0 C 0 a 0 a 1 C 1 C 1 Figure: Cut at these curves C 0,..., C n C i and C i a 0,..., a n : the maximal curves that do not divide the surface : the two sides after the cut : points on each curve

19 Elementary Contours S C 0 C 0 a 0 a 1 Γ 0 Γ 1 C 1 C 1 Figure: Cut at the curves Γ i as well Γ 0,..., Γ n : closed curves through the a i

20 Elementary Contours S C 0 C 0 a 0 a 1 Γ 0 Γ 1 b A C 1 C 1 Figure: Choose a point on each boundary curve b 0,..., b m : points on the boundary curves A 0...A m

21 Elementary Contours C 0 C 0 S P a 0 a 1 Γ 0 Γ 1 b A C 1 C 1 Figure: Choose a point P and connect it to the a i and b i P : point on the surface

22 Elementary Contours Three types of elementary contours : [P a i C i a i P ] [C i ] [P b i A i b i P ] [A i ] [P a i Γ i a i P ] [Γ i ]

23 Jordan s Idea Claim (Jordan) Every closed contour on the surface S is reducible to a unique sequence of elementary contours. In fact, he considers free deformations, so the sequence of elementary contours must allow cyclic permutations In hindsight, he has obtained a set of generators of π 1 S for S a genus n orientable surface with m boundary curves With the relation: [A 0 ]...[A m 1 ][C 0 ][Γ 0 ][C 0 ] 1 [Γ 0 ] 1...[C n 1 ][Γ n 1 ][C n 1 ] 1 [Γ n 1 ] 1 1

24 Why didn t he realize he wrote down generators for the fundamental group?

25 Why didn t he realize he wrote down generators for the fundamental group? He lacked: Continuous deformation of based loops (only requires the elementary contours to go through the point P ) Relation between elementary contours

26 Why didn t he realize he wrote down generators for the fundamental group? He lacked: Continuous deformation of based loops (only requires the elementary contours to go through the point P ) Relation between elementary contours The abstract group concept had not yet been formulated! It is not obvious how to interpret the fundamental group as a permutation group

27 A Classification Theorem Theorem (Jordan, 1866) Two orientable surfaces with boundaries are homeomorphic if and only if they have the same genus and the same number of boundary curves. His terminology for homeomorphic: applicable, one to the other without tearing or duplication

28 The Transition to Analysis Situs The theory of integration provides an abelian structure f = f + f = f α β α β β α

29 The Transition to Analysis Situs The theory of integration provides an abelian structure f = f + f = f α β α β β α But if we integrated a multi-valued function this might not be the case!

30 The Transition to Analysis Situs The theory of integration provides an abelian structure f = f + f = f α β α β β α But if we integrated a multi-valued function this might not be the case! The permutations of values of a multi-valued function along α β need not be the same as along β α

31 The Transition to Analysis Situs The theory of integration provides an abelian structure f = f + f = f α β α β β α But if we integrated a multi-valued function this might not be the case! The permutations of values of a multi-valued function along α β need not be the same as along β α Questions of underlying topology of a space began to be studied in their own right!

32 Poincaré and the Fundamental Group Buildup to the fundamental group : Consider unbranched multi-valued functions F i on a manifold If the functions are continued around a loop, they undergo a permutation He shows, permutations along closed paths on a manifold form a group

33 Poincaré and the Fundamental Group Buildup to the fundamental group : Consider unbranched multi-valued functions F i on a manifold If the functions are continued around a loop, they undergo a permutation He shows, permutations along closed paths on a manifold form a group Depends on the functions F i! Let G be group of all such functions

34 Poincaré and the Fundamental Group Introduces null-homotopic paths: If the point M describes an infinitely small contour on the manifold, the functions F will return to their original values. This remains true if M describes a loop on the manifold, that is to say, if it varies from M 0 to M 1 following an arbitrary path M 0 BM 1, then describes an infinitely small contour and returns from M 1 to M 0 traversing the same path M 1 BM 0.

35 Poincaré and the Fundamental Group Introduces null-homotopic paths: If the point M describes an infinitely small contour on the manifold, the functions F will return to their original values. This remains true if M describes a loop on the manifold, that is to say, if it varies from M 0 to M 1 following an arbitrary path M 0 BM 1, then describes an infinitely small contour and returns from M 1 to M 0 traversing the same path M 1 BM 0. Composition of paths transversed in opposite directions: M 0 AM 1 BM 0 + M 0 BM 1 CM 0 M 0 AM 1 CM 0

36 Poincaré and the Fundamental Group Introduces null-homotopic paths: If the point M describes an infinitely small contour on the manifold, the functions F will return to their original values. This remains true if M describes a loop on the manifold, that is to say, if it varies from M 0 to M 1 following an arbitrary path M 0 BM 1, then describes an infinitely small contour and returns from M 1 to M 0 traversing the same path M 1 BM 0. Composition of paths transversed in opposite directions: M 0 AM 1 BM 0 + M 0 BM 1 CM 0 M 0 AM 1 CM 0 Emphasizes that M 0 AM 1 CM 0 M 0 CM 1 AM 0

37 But... Even Poincaré gets some stuff wrong

38 But... Even Poincaré gets some stuff wrong M 0 BM 0 0 if the closed contour M 0 BM 0 constitutes the complete boundary of a 2-dimensional manifold contained in the the manifold.

39 But... Even Poincaré gets some stuff wrong M 0 BM 0 0 if the closed contour M 0 BM 0 constitutes the complete boundary of a 2-dimensional manifold contained in the the manifold. But he latter corrects it: K 0 mod V there is a simply connected region in V of which K is boundary

40 The Fundamental Group In this way, on can imagine a group G satisfying the following conditions: 1 For each closed contour M 0 BM 0 there is a corresponding substitution S of the group 2 S reduces to the identical substitution if and only M 0 BM If S and S correspond to the contours C and C and if C = C + C, the substitution corresponding to C will be SS This group G will be called the fundamental group of the manifold V.

41 Why Higher Homotopy Groups? Good question! Vienna, 1931 Cech presents a description of higher homotopy groups π n X := [S n, ; X, x 0 ] No applications for the homotopy groups Only one theorem

42 Why Higher Homotopy Groups? Good question! Vienna, 1931 Cech presents a description of higher homotopy groups π n X := [S n, ; X, x 0 ] No applications for the homotopy groups Only one theorem They were abelian (for n 2)

43 Why Higher Homotopy Groups? Persuaded by others that they could not possibly contain any additional information above the already known abelian homology groups Cech actually withdrew his paper π 1 X encompasses more information than its abelian counterpart H 1 (X) H 1 (X; Z) π 1 X/[π 1 X, π 1 X]

44 Why Higher Homotopy Groups? - In Hindsight There s interesting stuff going on: Hopf fibration Compare to H (S n ) = 0 for > n Action of the fundamental group on higher homotopy groups Eilenberg s obstruction theory

45 Hurewicz and a redefinition, 1935 Let Ω(X, x 0 ) be the loop space: set of based loops with basepoint S 1 X x 1 : S 1 {x 0 } and the compact-open topology. Definition/Proposition 1 Ω p (X, x 0 ) = Ω ( Ω p 1 (X, x 0 ), x p 1 ) 2 π p (X, x 0 ) = π 1 ( Ω p 1 (X, x 0 ), x p 1 )

46 An Action of π 1 on π n Given a path α : [0, 1] X with endpoints a, b X, induces an isomorphism π n (X, a) π n (X, b) Depends only on the homotopy class of the map. π 1 (X) acts by automorphisms on every π n (X, x 0 )

47 Homotopy Groups of Spheres Brouwer: 1912: degrees of maps on a sphere [f] degf Hopf: : maps of spheres into spheres...may righty be called the starting point of homotopy theory Dieudonné The Hopf fibration: S 3 p S 2 Hurewicz: 1935: π m S 1 = 0 for m 2 Freudenthal: 1937: homotopy suspension π i (S n ) π i+1 (S n+1 ) : i < 2n 1

48 Brouwer and Degree Let f : S n S n, then given by then, deg f = d f : H n (S n ) H n (S n ) α Z dα Brower s definition: Involves simplicial approximation and counting Theorem (Brouwer) The degree of a map is invariant under homotopy.

49 Brouwer and Degree At ICM 1912, conjectured that the converse was also true: Conjecture For M a connected, compact, orientable n-dimensional manifold, and f and g two continuous maps of the same degree, then f g. Published a proof for M = S 2 M S n Incredibly long, dense, intricate, and imprecise! 4 reductions to finally apply a result of Klein on compact Riemann surfaces of genus 0 Questionable arguments

50 Hopf and maps of spheres Proved Hopf s conjecture [f] degf π n (S n ) Z Introduced the Hopf invariant H(f) Used that to prove that π 3 (S 2 ) Z

51 The Hopf Fibration, 1931 S 1 S 3 p S 2 Direct construction: Identify R 4 with C 2 and R 3 with C R S 3 := {(z 0, z 1 ) : z z 1 2 = 1} S 2 := {(z, x) : z 2 + x 2 = 1} p(z 0, z 1 ) = (2z 0 z 1, z 0 2 z 1 2 )

52 The Hopf Fibration, 1931 S 1 S 3 p S 2 Direct construction: Identify R 4 with C 2 and R 3 with C R S 3 := {(z 0, z 1 ) : z z 1 2 = 1} S 2 := {(z, x) : z 2 + x 2 = 1} p(z 0, z 1 ) = (2z 0 z 1, z 0 2 z 1 2 ) Clearly maps to S 2, can check that p(z 0, z 1 ) = p(w 0, w 1 ) (w 0, w 1 ) = λ(z 0, z 1 ), λ = 1

53 Freudenthal and stable homotopy groups The based (reduced) suspension of X, ΣX is defined as ΣX = X [0, 1]/(X {0} X {1} {x 0 } [0, 1]) There is a natural bijection: [X, x 0 ; ΩY, y 1 ] (Σ and Ω are adjoint functors) [ΣX, x 0 ; Y, y 0 ]

54 Freudenthal and stable homotopy groups [X, x 0 ; ΩY, y 1 ] [ΣX, x 0 ; Y, y 0 ] We also have a natural map (Y, y 0 ) (ΩΣY, y 1 ) Putting the two together: In particular: [X, x 0 ; Y, y 0 ] [X, x 0 ; ΩΣY, y 1 ] E : [X, x 0 ; Y, y 0 ] [ΣX, x 0 ; ΣY, y 0 ] E : π r (S n ) π r+1 (S n+1 ) Freudenthal showed that this is an isomorphism for r < 2n 1 and surjective for r = 2n 1

55 Freudenthal and stable homotopy groups Which means: π r (S n ) E π r+1 (S n+1 E )... π r+k (S n+k )... Are isomorphisms for k > r 2n + 1

56 Freudenthal and stable homotopy groups Which means: π r (S n ) E π r+1 (S n+1 E )... π r+k (S n+k )... Are isomorphisms for k > r 2n + 1 The homotopy groups of spheres stabilize!

57 Samuel Eilenberg Figure: Eilenberg

58 Samuel Eilenberg Figure: Hmmm...?

59 Samuel Eilenberg Figure: Eilenberg

60 Eilenberg Bourbaki On homotopy groups and fibre spaces Eilenberg +MacLane Met in 1940 in Ann Arbor, wrote 15 papers together until 1954 Introduced category theory, the concepts of Hom, Ext, functor, natural transformation Cohomology of groups The relation between homotopy and homology Eilenberg + Steenrod Axiomatized homology and cohomology Eilenberg + Cartan Homological Algebra

61 Eilenberg Obstruction Theory Given a map X 1 Y If Y is simply connected this extends to a map on X 2. Null-homotopy in Y gives a way to fill in 2-skeleton. This generalizes: Have a map defined on X n, and (n + 1)-cell σ Attaching map g σ : S n X n f extends to σ if and only if f g σ is null-homotopic Ie if the cochain c(f) : σ [f g σ ] C n+1 (X; π n (Y )) is zero c(f): the obstruction cocycle The immediate result: this is always possible if π n (Y ) = 0

62 References History of Topology edited by I.M. James A Brief, Subjective History of Homology and Homotopy Theory in This Century by Peter Hilton A History of Algebraic and Differential Topology by Dieudonné