# Allen Back. Oct. 29, 2009

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1 Allen Back Oct. 29, 2009

2 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )).

3 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Sometimes the universal coefficients thm lets us go from these cases to coefficient ring Z.

4 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Smith (Annals of Math, 1938): Fixed point set of a Z p action on a homology n-sphere is a Z p homology r-sphere with r n.

5 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Smith (Annals of Math, 1938): Fixed point set of a Z p action on a homology n-sphere is a Z p homology r-sphere with r n. Smith did the Q case for toral actions too.

6 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Borel: (Anls Math Stds Smnr on Transformation Groups 60) Introduced the Borel construction (homotopy quotient) in the hope of doing Smith theory more systematically.

7 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Borel: (Anls Math Stds Smnr on Transformation Groups 60) Introduced the Borel construction (homotopy quotient) in the hope of doing Smith theory more systematically. Progress still took a while.

8 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Reformulation: ( 70) Organized around the philosophy that (for the above coeff. in the above grp cases) the free part of the equivariant cohomology goes most of the way towards determining the cohomology of the fixed point set. The torsion goes towards determining the orbit structure.

9 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Reformulation: ( 70) Organized around the philosophy that (for the above coeff. in the above grp cases) the free part of the equivariant cohomology goes most of the way towards determining the cohomology of the fixed point set. The torsion goes towards determining the orbit structure. For non-abelian Lie groups G this doesn t work. The Hsiangs already showed given any finite complex K, there is a compact fintie dimensional acylic G-space X on which G acts with fixed point set X G = K, Lowell Jones (Annals of Math 71) and Bob Oliver s work (Comm. Math. Helv 75) solidified this idea further.

10 [CTTTG] Cohomology Theory of Topological Transformation Groups,, 1975.

11 [CTTTG] Cohomology Theory of Topological Transformation Groups,, (Springer Ergebnisse der Mathematik Band 85) The bible, but sometimes a little terse.

12 [CTTTG] Cohomology Theory of Topological Transformation Groups,, (Springer Ergebnisse der Mathematik Band 85) The bible, but sometimes a little terse. For example the 17 pages of Chapter 2 are an almost complete development of the structure, classification, and rep theory of compact Lie groups.

13 [CTTTG] Cohomology Theory of Topological Transformation Groups,, (Springer Ergebnisse der Mathematik Band 85) The bible, but sometimes a little terse. [CMTG] Cohomological Methods in Transformation Groups, Chris Allday and Volker Puppe 1993.

14 [CTTTG] Cohomology Theory of Topological Transformation Groups,, (Springer Ergebnisse der Mathematik Band 85) The bible, but sometimes a little terse. [CMTG] Cohomological Methods in Transformation Groups, Chris Allday and Volker Puppe (Cambridge Studies in Advanced Mathematics 32) A nice recent book. Chris was Wu-Yi s first student after moving to Berkeley.

15 [CTTTG] Cohomology Theory of Topological Transformation Groups,, (Springer Ergebnisse der Mathematik Band 85) The bible, but sometimes a little terse. [CMTG] Cohomological Methods in Transformation Groups, Chris Allday and Volker Puppe [Brd] Introduction to Compact Transformation Groups, Glen Bredon 1972.

16 [CTTTG] Cohomology Theory of Topological Transformation Groups,, (Springer Ergebnisse der Mathematik Band 85) The bible, but sometimes a little terse. [CMTG] Cohomological Methods in Transformation Groups, Chris Allday and Volker Puppe [Brd] Introduction to Compact Transformation Groups, Glen Bredon Chapter 3 is a bare-handed approach to Smith theory.

17 Weights as Cohomology Classes Unless otherwise specified, the coefficient ring k is Q in the case of toral G = ( (S 1 ) n) actions and Z p in the case of G = Z p tori ( (Z p )) actions.

18 Weights as Cohomology Classes BS 1 = CP, B(Z 2 ) = RP. So for G a torus or Z 2 -tori, H (BG) = k[t 1,..., t r ] where deg t i is 2 or 1 respectively.

19 Weights as Cohomology Classes BS 1 = CP, B(Z 2 ) = RP. So for G a torus or Z 2 -tori, H (BG) = k[t 1,..., t r ] where deg t i is 2 or 1 respectively. For p odd, H (BG) = k[t 1,..., t r ] [ν 1,..., ν r ] where deg(t i ) = 2 and deg(ν i ) = 1.

20 Weights as Cohomology Classes BS 1 = CP, B(Z 2 ) = RP. So for G a torus or Z 2 -tori, H (BG) = k[t 1,..., t r ] where deg t i is 2 or 1 respectively. For p odd, H (BG) = k[t 1,..., t r ] [ν 1,..., ν r ] where deg(t i ) = 2 and deg(ν i ) = 1. In all these toral cases, for a (possibly p) torus of rank r, we use the notation R := k[t 1,..., t r ] for the polynomial part of the coefficient ring of equivariant cohomology H G and R 0 for its field of fractions.

21 Weights as Cohomology Classes A homomorphism H G induces a map BH BG and so a map H (BG) H (BH).

22 Weights as Cohomology Classes Thus a weight of a linear representation as a representation w : G S 1 is equally well viewed as a 2 dimensional cohomology class representing the pullback to BG of the generator in dim 2 of H (CP ). Also interpretable via transgression in the spectral sequence for G EG BG. Similarly in the p odd case of Z p, weights are 2 dimensional cohomology classes and in the Z 2 case 1 dimensional cohomology classes.

23 Wu-Yi s Reformulation [CTTTG page 47] R := k[t 1,..., t l ] is the polynomial part of H G (pt) = H (BG) and R 0 is its field of fractions.

24 Wu-Yi s Reformulation [CTTTG page 47] R := k[t 1,..., t l ] R 0 is its field of fractions. Let {ξ 1,..., ξ l, ν 1,..., ν m } be a generator system for the R 0 algebra H G (X, k) R R 0 with the ξ i of even degree and and the ν j of odd degree.

25 Wu-Yi s Reformulation [CTTTG page 47] R := k[t 1,..., t l ] R 0 is its field of fractions. Let {ξ 1,..., ξ l, ν 1,..., ν m } be a generator system F = F F s, F j connected, q j F j

26 Wu-Yi s Reformulation [CTTTG page 47] R := k[t 1,..., t l ] R 0 is its field of fractions. Let {ξ 1,..., ξ l, ν 1,..., ν m } be a generator system F = F F s, F j connected, q j F j I := kerρ : A = R 0 [t 1,..., t l ] R0 [ν 1,..., ν m ] H G (X, k) R 0.

27 Wu-Yi s Reformulation [CTTTG page 47] R := k[t 1,..., t l ] R 0 is its field of fractions. Let {ξ 1,..., ξ l, ν 1,..., ν m } be a generator system F = F F s, F j connected, q j F j I := kerρ : A = R 0 [t 1,..., t l ] R0 [ν 1,..., ν m ] H G (X, k) R 0. Then the radical of I decomposes I = M1 M 2... M s where the M j are maximal ideals M(α j ) with α j = (α j1,..., α jl ) R l 0.

28 Wu-Yi s Reformulation [CTTTG page 47] R := k[t 1,..., t l ] R 0 is its field of fractions. Let {ξ 1,..., ξ l, ν 1,..., ν m } be a generator system F = F F s, F j connected, q j F j So the variety V (I ) = {α 1, α 2,..., α s }.

29 Wu-Yi s Reformulation [CTTTG page 47] R := k[t 1,..., t l ] R 0 is its field of fractions. Let {ξ 1,..., ξ l, ν 1,..., ν m } be a generator system F = F F s, F j connected, q j F j So the variety V (I ) = {α 1, α 2,..., α s }. Components F j of the fixed point set are in 1:1 correspondence with the α j so that ι : q j F j X satisfies ι j ξ i = α ji.

30 Wu-Yi s Reformulation [CTTTG page 47] R := k[t 1,..., t l ] R 0 is its field of fractions. Let {ξ 1,..., ξ l, ν 1,..., ν m } be a generator system F = F F s, F j connected, q j F j So the variety V (I ) = {α 1, α 2,..., α s }. Components F j of the fixed point set are in 1:1 correspondence with the α j so that ι : q j F j X satisfies ι j ξ i = α ji. H (F j, k) k R 0 = A/Ij where I j = I Mj A and I Mj is the localization of I at M j.

31 Wu-Yi s Reformulation [CTTTG page 47] R := k[t 1,..., t l ] R 0 is its field of fractions. Let {ξ 1,..., ξ l, ν 1,..., ν m } be a generator system F = F F s, F j connected, q j F j So the variety V (I ) = {α 1, α 2,..., α s }. Components F j of the fixed point set are in 1:1 correspondence with the α j so that ι : q j F j X satisfies ι j ξ i = α ji. I = I 1 I 2... I s = I 1 I 2... I s

32 Borel Criterion View X X G BG more explicitly as X X G EG G\EG where elements of X G are [x, e] with x X, e EG, and [x, e] [xg, g 1 e].

33 Borel Criterion View X X G BG more explicitly as X X G EG G\EG where elements of X G are [x, e] with x X, e EG, and [x, e] [xg, g 1 e]. View elements of BG as cosets Ge with e EG.

34 Borel Criterion View X X G BG more explicitly as X X G EG G\EG where elements of X G are [x, e] with x X, e EG, and [x, e] [xg, g 1 e]. View elements of BG as cosets Ge with e EG. If the fixed point set is nonempty (say q F ), then the map Ge [q, g] gives a section of X X G EG G\EG.

35 Borel Criterion View X X G BG more explicitly as X X G EG G\EG where elements of X G are [x, e] with x X, e EG, and [x, e] [xg, g 1 e]. View elements of BG as cosets Ge with e EG. If the fixed point set is nonempty (say q F ), then the map Ge [q, g] gives a section of X X G EG G\EG. Borel Criterion: The fixed point set F is nonempty iff H (BG) H (X G ) is injective.

36 Serre spectral sequence applied to X X G BG shows (So this is the I trivial case.) HG (X ) = R.

37 Serre spectral sequence applied to X X G BG shows (So this is the I trivial case.) HG (X ) = R. HG (X ) = R and H (F j, k) k R 0 = R/IMj means F j must be acyclic.

38 Sphere Sps. X is a cohomology sphere with k coefficients. The E 2 term of the spectral sequence for X X G BG is then R H (X ).

39 Sphere Sps. X is a cohomology sphere with k coefficients. The E 2 term of the spectral sequence for X X G BG is then R H (X ). Because the spectral sequence has only 2 rows, there is only one possible nonzero differential. And H (BG) H (X G ) is injective iff the spectral sequence collapses.

40 Sphere Sps. X is a cohomology sphere with k coefficients. The E 2 term of the spectral sequence for X X G BG is then R H (X ). Because the spectral sequence has only 2 rows, there is only one possible nonzero differential. And H (BG) H (X G ) is injective iff the spectral sequence collapses. So by the Borel criterion, there is a nonempty fixed point set exactly when the spectral sequence collapses at the E 2 stage.

41 Sphere Sps. X is a cohomology sphere with k coefficients. The E 2 term of the spectral sequence for X X G BG is then R H (X ). Let x and 1 be the lifts of the generators of H (X ) to H (X G ).

42 Sphere Sps. X is a cohomology sphere with k coefficients. The E 2 term of the spectral sequence for X X G BG is then R H (X ). Let x and 1 be the lifts of the generators of H (X ) to H (X G ). When F φ, H G (X ) R R 0 = R0 [x]/(f (x)) where f (x) = x 2 + ax + b is the quadratic polynomial expressing x 2 in terms of our module basis {1, x} of HG (X ) as an R-module.

43 Sphere Sps. X is a cohomology sphere with k coefficients. The E 2 term of the spectral sequence for X X G BG is then R H (X ). Let x and 1 be the lifts of the generators of H (X ) to H (X G ). When F φ, H G (X ) R R 0 = R0 [x]/(f (x)) By the rationality part of Wu-Yi s fundamental fixed point theorem, f (x) = (x α 1 )(x α 2 ) in R 0 [x].

44 Sphere Sps. X is a cohomology sphere with k coefficients. The E 2 term of the spectral sequence for X X G BG is then R H (X ). Let x and 1 be the lifts of the generators of H (X ) to H (X G ). When F φ, H G (X ) R R 0 = R0 [x]/(f (x)) Localizing at the maximal ideals tells us F is two points (S 0 ) if α 1 α 2 and S r if α 1 = α 2

45 Sphere Sps. X is a cohomology sphere with k coefficients. The E 2 term of the spectral sequence for X X G BG is then R H (X ). Let x and 1 be the lifts of the generators of H (X ) to H (X G ). When F φ, H G (X ) R R 0 = R0 [x]/(f (x)) This is the original Smith theorem.

46 Sphere Sps. X is a cohomology sphere with k coefficients. The E 2 term of the spectral sequence for X X G BG is then R H (X ). Let x and 1 be the lifts of the generators of H (X ) to H (X G ). When F φ, H G (X ) R R 0 = R0 [x]/(f (x)) Note it is really the multiplicative structure of the cohomology ring which distinguishes examples.

47

48 Consider a torus action on a rational cohomology CP n.

49 Consider a torus action on a rational cohomology CP n. Analogous results exist for Z p tori and other projective spaces, but they are more difficult to establish.

50 Consider a torus action on a rational cohomology CP n. For parity reasons, the spectral sequence of X X G BG collapses and H G (X ) = H (X ) k R as a module. (Also the Leray-Hirsch thm)

51 Consider a torus action on a rational cohomology CP n. For parity reasons, the spectral sequence of X X G BG collapses and H G (X ) = H (X ) k R as a module. (Also the Leray-Hirsch thm) Let x be the lift of a generator of H 2 (X ) to H 2 (X G ). Then as a ring H (X G ) = R[x]/(f (x)) where f (x) is a monic polynomial of degree n.

52 Consider a torus action on a rational cohomology CP n. For parity reasons, the spectral sequence of X X G BG collapses and H G (X ) = H (X ) k R as a module. (Also the Leray-Hirsch thm) By the rationality part of Wu-Yi s fundamental fixed point theorem, H (X G ) R R 0 = R 0 [x]/(f (x)) where f (x) = Π(x α i ) in R 0 [x].

53 Consider a torus action on a rational cohomology CP n. For parity reasons, the spectral sequence of X X G BG collapses and H G (X ) = H (X ) k R as a module. (Also the Leray-Hirsch thm) By the Gauss lemma, f (x) already splits into linear factors over R. f (x) = Π i = 1 s (x α i ) m i

54 Consider a torus action on a rational cohomology CP n. For parity reasons, the spectral sequence of X X G BG collapses and H G (X ) = H (X ) k R as a module. (Also the Leray-Hirsch thm) Localization at the maximal ideals then gives s fixed point components, each a cohomology CP m i 1.

55 Involutions on My Ph. D. thesis. Was once accepted subject to revision for the Memoirs but the editor wanted a less computational proof which I have never known how to do.

56 Involutions on Using the splitting principle, view where the t i are of degree 1. H (G 2 (R N ), Z 2 ) Z 2 [t 1, t 2 ].

57 Involutions on Let h i be the i th complete symmetric function of t 1, t 2. So h k = s k+1 /s 1 where s k is the k th power sum, t1 k + tk 2 in this case.

58 Involutions on Let h i be the i th complete symmetric function of t 1, t 2. So h k = s k+1 /s 1 where s k is the k th power sum, t1 k + tk 2 in this case. The h k are the Stiefel-Whitney classes of the canonical N 1 plane bundle over G 2 (R N ).

59 Involutions on Prop: H (G 2 (R N ), Z 2 ) = Z 2 [h 1, h 2 ]/(h N 1, h N ). {h a h b : 0 a N 2, 0 b N 2} form a module basis for this ring. h a h b h c = Σ a 1 j=0 h jh a+b+c j + Σ a j=0h b+j h a+c j even over Z.

60 Involutions on Linear models: Linear representations of G = Z 2 into Gl(N) are indexed by the number l of 1 s. The two canonical bundles over X = G 2 (R N ) are equivariant with respect to such an action, so for a linear action, the spectral sequence of X X G BG collapses. Hence for a linear action and it is not hard to show that H (X G ) = R[h 1, h 2 ]/(f N 1, f N ) f N k = Σ N k j=0 ρ jh N k j where ρ j is the j th elementary symmetric function of the weights (counting multiplicities) of the Z 2 representation.

61 Involutions on Based on Wu-Yi s version of the fixed point theorem: Prop: If an involution has the same equivariant cohomology as a linear model, then cohomologically its fixed point sets are also the same.

62 Involutions on Based on Wu-Yi s version of the fixed point theorem: Prop: If an involution has the same equivariant cohomology as a linear model, then cohomologically its fixed point sets are also the same. In general we have three components, G 2 (R l ), RP l 1 RP N l 1, and G 2 (R N l ).

63 Involutions on For a general involution, consider the Serre spectral sequence for X X G BG. Prop: If N is even and not congruent to 64 mod 192, then the action of π 1 (BG) on H (X, Z 2 ) is trivial. For N even, if there is a nonempty fixed point set, then the spectral sequence collapses at the E 2 level.

64 Involutions on Thus under the above dimension restrictions, for any involution, if F φ, we have fairly quickly shown that H G (X ) = Z 2[h 1, h 2, x]/(f, g) where f = h N 1 (mod x) and g = h N (mod x).

65 Involutions on 90% of the work in my thesis is using the Steenrod algebra invariance of (f, g) to show that f and g must be the same polynomials as for one of the linear models.

66 Involutions on This gives the theorem that cohomologically (for these values of N), any involution on G 2 (R N ) has fixed point set with the same Z 2 cohomology as one of the linear models.

67 Involutions on This gives the theorem that cohomologically (for these values of N), any involution on G 2 (R N ) has fixed point set with the same Z 2 cohomology as one of the linear models. There is a nice part of the Steenrod algebra argument that works generically and only takes perhaps 20 pages to write down.

68 Involutions on This gives the theorem that cohomologically (for these values of N), any involution on G 2 (R N ) has fixed point set with the same Z 2 cohomology as one of the linear models. There is a nice part of the Steenrod algebra argument that works generically and only takes perhaps 20 pages to write down. It is based on showing that a certain mod 2 binomial coefficient is usually nonzero.

69 Involutions on This gives the theorem that cohomologically (for these values of N), any involution on G 2 (R N ) has fixed point set with the same Z 2 cohomology as one of the linear models. There is a nice part of the Steenrod algebra argument that works generically and only takes perhaps 20 pages to write down. It is based on showing that a certain mod 2 binomial coefficient is usually nonzero. Unfortunately, that coefficient does sometimes vanish, so there are a lot of special cases that had to be checked to give the full theorem.

70 Involutions on This gives the theorem that cohomologically (for these values of N), any involution on G 2 (R N ) has fixed point set with the same Z 2 cohomology as one of the linear models. There is a nice part of the Steenrod algebra argument that works generically and only takes perhaps 20 pages to write down. It is based on showing that a certain mod 2 binomial coefficient is usually nonzero. Probably this example is one of the most computationally intricate examples of the fixed point theorem that has ever been written down.

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