Celebrating One Hundred Fifty Years of. Topology. ARBEITSTAGUNG Bonn, May 22, 2013

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1 Celebrating One Hundred Fifty Years of Topology John Milnor Institute for Mathematical Sciences Stony Brook University ( ARBEITSTAGUNG Bonn, May 22, 2013

2 Algebra & Number Theory Geometry

3 3 Algebra & Number Theory 5 4 Geometry Theorema Egregium iπ e = 1 Analysis

4 3 Algebra & Number Theory 5 4 Geometry TOPOLOGY Theorema Egregium iπ e = 1 Analysis

5 When did topology start? V E +F =

6 W C (p) = 1 2πi C dz z p Cauchy, 1825

7 1 L = 4π ZZ x,y (x y ) (dx dy ) kx y k3 1833

8 doubly connected triply connected Riemann, 1857 triply connected

9 August Ferdinand Möbius

10 The Möbius Classification of surfaces in R 3 : 1863 Definition of the class of a surface: On a closed surface of the n-th class [= genus n 1], there exist n 1 closed curves which do not disconnect the surface. Theorem. Any two closed surfaces of the same class are elementarily related. Two geometric figures will be called elementarily related if to any infinitely small element of any dimension in one figure there corresponds an infinitely small element in the other figure, such that two neighboring elements in one figure correspond to two elements in the other which also come together;

11 Camille Jordan, 1877, Jordan curve theorem. Walther von Dyck, 1888: Topology studies properties invariant under continuous functions with continuous inverse. χ(m) = 1 K da. 2π Henri Poincaré, , homology, Betti numbers, duality, homotopy, fundamental group, covering spaces,

12 20 th century Felix Hausdorff L. E. J. Brouwer H. Kneser and H. Hopf

13 Solomon Lefschetz and James Alexander

14 The Alexander Chimney in Colorado

15 Hassler Whitney E Sn 1 X characteristic classes w i H i (X; Z/2).

16 Lev Pontryagin ξ real vector bundle over X p j (ξ) H 4j (X; Z) Shiing-Shen Chern γ complex vector bundle over X c j (γ) H 2j (X; Z)

17 Many people put these into more modern form Wu Wen-Tsün H (B GL(R) ; Z/2) = (Z/2)[w1, w 2, w 3, ] H (B GL(C) ; Z) = Z[c1, c 2, c 3, ] H (B GL(R) ; Z) = Z[p1, p 2, ] (2 torsion)

18 Neue topologische Methoden

19 Todd genus ( arithmetic genus) Francesco Severi David Hilbert Lemma. a unique a'j7" J. A. Todd T = c (c c 2) c 1c 2 + H (B GL(C) ; Q), such that the genus T (V n ) = T(τ V n )[V n] is multiplicative: T (V V ) = T (V ) T (V ), with T ( P n (C) ) = +1. Theorem. T (V n ) = n ( 1) k dim C {holomorphic k forms}. k=0

20 Some notation If γ is a holomorphic vector bundle over V, let (γ) denote the sheaf of germs of local holomorphic sections. Let 1 denote the trivial line bundle. Pierre Dolbeault: Theorem : H k V ; (1) = {holomorphic k forms}. P k k Hence T (V ) = k ( 1) dimc H V ; (1).

21 Classical Riemann-Roch Theorems Gustav Roch Max Noether Andre Weil

22 The Chern character of a complex n-plane bundle over X ch(γ n ) = n + c 1 1! + c 1 2 2c 2 2! + c 3 1 3c 1c 2 + 3c 3 3! + is an element of H (X; Q) characterized by two properties: ch(γ 1 ) = e c 1(γ 1 ). ch(γ m γ n) = ch(γ m ) + ch(γ n), = ch(γ m γ n) = ch(γ m ) ch(γ n). Hirzebruch s Riemann-Roch Theorem: For any holomorphic vector bundle γ over V, ( 1) k dim C H k( V ; (γ) ) ( ) = ch(γ) T(τ V ) [V ]. k

23 Rene Thom s cobordism theory was based on deep geometric intuition, plus hard algebraic topology. Essential ingredients: Jean-Pierre Serre on spectral sequences Norman Steenrod on cohomology operations

24 The L-genus of an oriented 4n-manifold Hirzebruch showed that there is one and only one sum L = 1 + p p 2 p H (B GL(R) ; Q), such that the L-genus, L(M 4n ) = L(τ M 4n)[M 4n ], is multiplicative L(M M ) = L(M) L(M ), with L ( P 2n (C) ) = +1. Theorem (Thom, Hirzebruch). The to the signature of the quadratic form L-genus L(M 4k ) is equal H 2k (M 4k ; Q) Q x (x x)[m 4k ].

25 Hirzebruch also defined the Â-genus Â(M4k ), where  = 1 p p 1 2 4p If M 4k is a spin manifold (w 2 = 0), then Michael Atiyah Is Singer proved that Â(M 4k ) is equal to the index of the associated Dirac operator, and hence is an integer.

26 Almost parallelizable manifolds Suppose that M 4k (point) is parallelizable, so that p j (τ M 4k ) is zero for j < k. Hirzebruch s formulas then take the form L(M 4k ) = and since w 2 = 0, ( 2 2k (2 2k 1 1) B ) k p k [M 4k ], (2k)! Â(M 4k ) = B k 2 (2k)! p k[m 4k ].

27 Raoul Bott at the Arbeitstagung in 1969 Bott showed that π 4k 1 (SO) = Z. Furthermore, a generator gives rise to a vector bundle ξ over S 4k with { p k (ξ)[s 4k (2k 1)! for k even ] = 2(2k 1)! for k odd. = the Pontrjagin number p k [M 4k ] of an almost parallelizable manifold is always divisible by (2k 1)!.

28 The Pontrjagin-Thom construction. S p+q f S q S p (or M p ) = f -1 (x 0 ) x 0 Any M p S p+q with framed normal bundle determines a homotopy class in π p+q (S q ). Taking M p = S p we obtain the J-homomorphism J : π p (SO q ) π p+q (S q ). In the stable case q >> p, I will write J : π p (SO) Π p.

29 The Adams Conjecture Frank Adams J ( π 4k 1 (SO) ) ( ) Bk = denominator 4k.

30 The E 8 manifold-with-boundary W 4k The boundary W 4k is a homotopy (4k 1)-sphere if k > 1.

31 Michel Kervaire Theorem The group of homotopy spheres which bound parallelizable manifolds is cyclic of order ( ) 2 2k 2 (2 2k 1 4Bk 1) numerator, k with generator W 4k 1.

32 The last 50 years Amazing progress in low dimensional topology: Freedman, Donaldson, Thurston Geometrization, Perelman Ever deeper connections with mathematical physics: gauge theory, Seiberg-Witten theory, symplectic topology, TO BE CONTINUED!

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