1. COHOMOLOGY WITH & Sunil Chebolu Illinois State University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A ISMAA Annual meeting, Bradley University, April 3-4 2009.

2. Plan of my talk  Motivation  Cohomology  Cohomology with stones and sticks  Applications 1.Groups acting on spheres 2.Generating hypothesis

3. Plan of my talk  Motivation  Cohomology  Cohomology with stones and sticks  Applications 1.Groups acting on spheres 2.Generating hypothesis

4. Motivation k - Field of characteristic p (p > 0 prime) G - Finite group kG - Group ring = {  g 2 G α g g | α g 2 k } Ultimate Goal: Explore the hidden secrets of finite groups. A traditional approach: One studies finite groups by examining how they act on k - vector spaces.

5. A simple example C 2 = h ¾ | ¾ 2 = 1 i acts on k 2 = k © k ¾ (v) = -v ¾ 2 (v) = ¾ ( ¾ (v)) = ¾ (-v) = -(-v) = v Therefore ¾ 2 = 1 This is a 2-dimensional representation of C 2 V -V

6. Definition: An n-dimensional representation of G is a homomorphism of groups ½ : G ! GL n (k). GL n (k) = group of all invertible n £ n matrices over k. Example: C 2 ! GL 2 (k) ¾  – I 2 1  I 2

7. Thus, an n-dimensional representation of G is an n-dimensional vector space V over k on which G acts by linear automorphisms. || n-dimensional kG - module Fundamental problem: Given G, classify all of its representations. This problem is notoriously hard !

8. To study finite groups and their representations, a very powerful algebraic machine was introduced in the early 20 th century. This is Group Cohomology --- one of the biggest inventions in the 20 th century. The cohomology ring R G is a very useful object to study invariants of G G finite group + R G cohomology ring +, x

9. Henri Poincaré

10. David Hilbert

11. The history of cohomology is long and rich – Early 1900s: Inception – Late 1940s: Matured – Thereafter: active area of research Three avatars of cohomology –Representation theory (G = finite group) Group cohomology –Topology (G = topological group) Continuous cohomology –Number theory (G = Galois group) Galois cohomology

12. Plan of my talk  Motivation  Cohomology  Cohomology with stones and sticks  Applications 1.Groups acting on spheres 2.Generating hypothesis

13. Cohomology R G = H * (G, k) = © n ≥ 0 H n (G,k) is a ring. char(k) divides |G|  Modular case Towards the definition of cohomology: kG linear map: This is a map Á : M ! N between representations M and N – Á is a linear transformation – Á (g ¢ m) = g Á (m) for all g in G and m in M. Hom G (M, N) = all kG-linear maps from M to N.

14. Examples of representations –Trivial representation of G is k with trivial G action, i.e., gx = x –Regular representation of G is kG kG = {  g 2 G α g g | α g 2 k } –Free representation of G is kG © kG ©  © kG –Sygizies of a representation M are   M = ker( F 1 ³ M)   M = ker( F 2 ³   M)  n M = ker( F n ³  n-1 M) for n > 1.

15. Definition of H n (G,M) H n (G,M) = Hom G (  n k, M)/~ f,g :  n k ! M are homotopic (f~g) iff f – g:  n k M F H * (G, M) = © n ≥ 0 H n (G,M)  graded k-vector space. Notation: Hom(A, B) = Hom G (A, B)/~

16. Important special case: M = k (trivial rep.) H * (G, k) = © n ≥ 0 H n (G,k)  graded ring Ring structure: ® 2 H 2 (G, k) ¯ 2 H 3 (G, k)   k k   k k ®¯ :   k   k =   k k ) ®¯ 2 H 5 (G,k) ® ¯ 3®3® ¯

17. Plan of my talk  Motivation  Cohomology  Cohomology with stones and sticks  Applications 1.Groups acting on spheres 2.Generating hypothesis

18. Cohomology with stones and sticks C 2 = h ¾ | ¾ 2 = 1 i Char(k) = 2 Freshman’s dream kC 2 = k[ ¾ ]/( ¾ 2 – 1) = k[ ¾ ]/( ¾ -1) 2 = k[x]/(x 2 ) k = kC 2 =  k = ker( kC 2 ! k) = ker = = k )  i k = k for all i. x

19. H i (C 2, k) = Hom(  i k, k) = Hom(k, k) = k h x i i H * (C 2, k) = © i≥0 k h x i i Products: x 1 x 1 = x 1 2 2 H 2 (C 2,k) = h x 2 i   k k   k k x 2 : k k = k k x 1 2 = x 2 Similarly, x 1 n = x n. ) H * (C 2, k)  k[x 1 ] x1x1 x1x1 x1x1 x1x1

20. Klein four group V 4 V 4 = C 2 © C 2 = h ¾, ¿ | ¾ 2 = 1 = ¿ 2 = 1, ¾ ¿ = ¿ ¾ i This is our favourite group ! kV 4 = k[x, y]/(x 2, y 2 ) x = ¾ -1, y = ¿ -1 = k h 1, x, y, xy i k =kV 4 = x x y y

21. Sygizies of k   k = ker (kV 4 ³ k) = ker =   k = ker (kV 4 © kV 4 ³   k) = ker © =

22. Similarly, we have  n k =   -n k =  n copies

23. Computing H * (V 4, k) H 1 (V 4, k) = Hom(   k, k) = Hom, = k © k = k h u i © k h v i u v

24. H 2 (V 4, k) = Hom(   k, k) = Hom, = k © k © = k h l i © k h m i © k h r i l r m

25. Products Educated guess: l = u 2, m = uv, r = v 2 ! u 2 = l u.u = = = uu u2u2 u

26. uv = m u.v = = = u v uv v

27. v 2 = r v v = = = vv v2v2 v

28. H n (V 4, k) = Hom(  n k, k) = Hom , = k-span h u n, u n-1 v, , uv n-1, v n i. Combining all these,we have: H * (V 4, k) = © H n (V 4, k) = © k-span h u n, u n-1 v, , uv n-1, v n i = k [u, v]

29. Plan of my talk  Motivation  Cohomology  Cohomology with stones and sticks  Applications 1.Groups acting on spheres 2.Generating hypothesis

30. Applications §1.Groups acting freely on spheres. Problem: Given a finite group G, can it act freely on some sphere (S n, for some n). Definition: An action Á : G £ S n ! S n is free if Á (g, x) = x ) g = e. Theorem: If G acts freely on S n then H * (G,k) is periodic with period dividing n.

31. Special case: G = V 4 Recall: H * (V 4, k)  k[u, v]. In particular, H n (V 4, k) is a k-vector space of dimension n+1 with basis { u n, u n-1 v, , v n }. ) lim n ! 1 dim k H n (V 4, k) = lim n ! 1 n+1 = 1 Conclusion: V 4 cannot act freely on any sphere. !!! YAY !!!

32. §2. Generating hypothesis -- joint work with Carlson and Mináč. Jan Mináč Jon Carlson

33. G finite group and M, N kG –modules Á : M ! N be a kG-linear map. Á induces a map in cohomology: H i (G, Á ) : H i (G, M) ! H i (G, N) ®  Á ± ® Á ± ® :  i k M N ® Á Induced map in cohomology

34. The problem of generating hypothesis Suppose H i (G, Á ) : H i (G, M) ! H i (G, N) is 0 for all i>0. Does this imply that Á : M ! N is null-homotopic? i.e., does Á factor through a free representation? Remark: The converse of the above problem is true for a trivial reason. ) H i (G, Á ) = 0 because H i (G, F) = 0 8 i > 0 M N F Á

35. Theorem: (Carlson, C, Mináč) Generating hypothesis holds for G The p-Sylow subgroup of G = C 2 or C 3

36. h is not null homotopic A map h of kV 4 modules

37. h is zero in cohomology

38. Ars Longa Vita Brevis Thank you