Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups
aa r X i v : . [ m a t h . A T ] A ug CHROMATIC FIXED POINT THEORY AND THEBALMER SPECTRUM FOR EXTRASPECIAL 2-GROUPS.
NICHOLAS J. KUHN AND CHRISTOPHER J. R. LLOYD
Abstract.
In the early 1940’s, P.A.Smith showed that if a finite p –group G acts on a finite complex X that is mod p acyclic, then its spaceof fixed points, X G , will also be mod p acyclic.In their recent study of the Balmer spectrum of equivariant stablehomotopy theory, Balmer and Sanders were led to study chromatic ver-sions of this statement, with the question: given H < G and n , what isthe smallest r such that if X H is acyclic in the ( n + r )th Morava K –theory, then X G must be acyclic in the n th Morava K –theory? Barthelet. al. then answered this when G is abelian, by finding general lower andupper bounds for these ‘blue shift’ numbers which agree in the abeliancase.In our paper, we first prove that these potential chromatic versionsof Smith’s theorem are equivalent to chromatic versions of a 1952 the-orem of E.E.Floyd, which replaces acyclicity by bounds on dimensionsof homology, and thus applies to all finite G –spaces. This unlocks newtechniques and applications in chromatic fixed point theory.In one direction, we are able to use classic constructions and repre-sentation theory to search for blue shift number lower bounds. We givea simple new proof of the known lower bound theorem, and then getthe first results about nonabelian 2-groups that don’t follow from pre-viously known results. In particular, we are able to determine all blueshift numbers for extraspecial 2-groups.As samples of new applications, we offer a new result about involu-tions on the 5-dimensional Wu manifold, and a calculation of the mod2 K -theory of a 100 dimensional real Grassmanian that uses a C chro-matic Smith theorem. Introduction If G is a finite group, say that G –space X is a finite G –space if it is aretract of a finite G –CW complex in the G –equivariant homotopy category.We let X G denote its subspace of fixed points.With a series of papers beginning with [S38], P.A.Smith used homologicalmethods to study the structure of the fixed points of such finite G –spaces,when G is a p –group. In particular, [S41, Theorem II] shows the following: Date : July 31, 2020.2010
Mathematics Subject Classification.
Primary 55M35; Secondary 55N20, 55P42,55P91, 57S17.
Let H be a subgroup of a finite p –group G , and let X be a finite G –space.If e H ∗ ( X H ; Z /p ) = 0 then e H ∗ ( X G ; Z /p ) = 0 . Note that, since a p –group is solvable, this theorem for all H < G is aconsequence of the special case { e } < C p .A decade later, E.E.Floyd upgraded this result to one that gives informa-tion about all finite G –spaces, not just those that are Z /p –acyclic. In ourcontext, [F52, Theorem 4.4] says: Let H be a subgroup of a finite p –group G , and let X be a finite G –space.Then dim Z /p H ∗ ( X H ; Z /p ) ≥ dim Z /p H ∗ ( X G ; Z /p ) . Recently [BS17, 6A19] the study of chromatic versions of Smith’s theoremhave arisen in the context of work on the Balmer spectrum for the G –equivariant stable homotopy category.We explain what these theorems would be. Recall that, for each prime p , the Morava K-theories are a family of generalized homology theoriesequipped with products. K (0) ∗ ( X ) = H ∗ ( X ; Q ) and, for n ≥
1, the theory K ( n ) ∗ ( X ) has coefficient ring K ( n ) ∗ equal to the graded field Z /p [ v ± n ], with | v n | = 2 p n −
2, and satisfies a Kunneth theorem. It is sometimes convenientto let K ( ∞ ) ∗ ( X ) = H ∗ ( X ; Z /p ).In [Rav84, Theorem 2.11], Ravenel proved the following acyclicity theo-rem: given m < ∞ , if a finite X is K ( m ) ∗ –acyclic then X is K ( n ) ∗ –acyclicfor all n < m . Atiyah–Hirzebruch spectral sequence considerations showthat this holds also for m = ∞ .We then have the following problem. Problem 1.1.
Given a finite p –group G , and subgroup H , for what pairs( m, n ) is it true that for all finite G –spaces X ,whenever e K ( m ) ∗ ( X H ) = 0 then e K ( n ) ∗ ( X G ) = 0?When this is true, we will say that the ( G, H, n, m ) Chromatic Smith Theo-rem holds.Note that Ravenel’s theorem answers this question when G is the trivialgroup, so this problem can be regarded as the search for the correct commongeneralization of Ravenel’s and Smith’s theorems.As we will review in §
4, given a finite p –group G , answering this problemfor all ( H, K, n, m ) with
K < H < G is equivalent to: • identifying the inclusions between prime ideals in the the Balmerspectrum for the G –equivariant stable homotopy category, • characterizing which ‘chromatic type functions’ can be realized byfinite G –spaces or spectra.The paper [BS17] made clear that this is a deep and interesting area tostudy, organized by the need to compute ‘blue shift’ numbers. Then in HROMATIC FIXED POINT THEORY 3 [6A19], Problem 1.1 was solved when G is an abelian p –group, by findinggeneral group theoretic upper and lower bounds for blue shift numbers whichagree when G is abelian.In our paper, we first show that chromatic Smith theorems are equivalentto chromatic versions of Floyd’s theorem. Our proof uses an old idea of JeffSmith which ultimately relies on the modular representation theory of thesymmetric groups.This reformulation allows us to use classic constructions and (ordinary)representation theory to search for examples giving blue shift number lowerbounds. We give a simple new proof of the lower bound theorem of [6A19],and then get the first results about nonabelian groups that don’t follow frompreviously known results. In particular, we are able to answer Problem 1.1for a family of 2-groups which include all extraspecial 2-groups.One can also apply the chromatic Floyd theorems in situations where thecorresponding chromatic Smith theorem was already known. We will give ajust a taste of this with a couple of explicit examples - a new result aboutinvolutions on the 5-dimensional Wu manifold, and a calculation of the mod2 K -theory of the real Grassmanian Gr ( R ) that uses a C chromaticSmith theorem to control the differentials in the nonequivariant Atiyah-Hirzebruch spectral sequence. Remark . Problem 1.1 is really a problem about equivalence classes ofpairs (
G, H ) with
H < G , where ( G , H ) ∼ ( G , H ) if there exists an iso-morphism α : G ∼ −→ G such that α ( H ) = H . Similarly other statementsin this paper are statements about such equivalence classes.In the next section, we will describe our main results in detail.1.1. Acknowledgements.
Our example resolving the open question when G = D was first found in June, 2019, with a computer search by the secondauthor using GAP. The second exceptional 2–group example, with G oforder 32, was also found this way, which led to us discovering this infinitefamily of examples. Tim Dokchister’s lovely website GroupNames has beenuseful.The D example was presented by the first author in a talk in Oberwolfachin August, 2019 [KL19]. At the same meeting, Markus Hausmann asked ifa chromatic Floyd theorem might be true, and it eventually occurred to usthat we had been working for awhile with an answer to this question, leadingto our presentation here.The first author is a PI of RTG NSF grant DMS-1839968, which haspartially supported the research of the second author.2. Main Results
We now describe our main results in detail. It will be useful to introducesome notation: let k n ( X ) = dim K ( n ) ∗ K ( n ) ∗ ( X ), when X is a finite complex. KUHN AND LLOYD
The numbers k ( X ) , k ( X ) , k ( X ) , . . . form a nondecreasing sequence – seeRemark 2.3(b) below – which stablizes at k ∞ ( X ) = dim Z /p H ∗ ( X ; Z /p ).2.1. Chromatic Floyd theorems.
The analogues of Floyd’s theorem goas follows.
Problem 2.1.
Given a finite p –group G , and subgroup H , for what pairs( m, n ) is it true that for all finite G –spaces X , k m ( X H ) ≥ k n ( X G )?When this is true, we will say that the ( G, H, n, m ) Chromatic Floyd Theo-rem holds.The (
G, H, n, m ) Chromatic Floyd Theorem clearly implies the (
G, H, n, m )Chromatic Smith Theorem. Our perhaps surprising discovery is that theyare, in fact, equivalent.
Theorem 2.2.
If the ( G, H, n, m ) Chromatic Smith Theorem is true thenthe ( G, H, n, m ) Chromatic Floyd Theorem is true. In §
6, we prove this theorem in its contrapositive form. We look backwardsto move forwards: we use, in our equivariant setting, a construction used inthe mid 1980’s by Jeff Smith in the proof of the Periodicity Theorem [HS98].This lets us show that, if a based finite G –space X satisfies k m ( X H )
Ravenel’s acyclicity thoerem implies that if the(
G, H, n, m ) Chromatic Smith Theorem is true, then it is true for all pairs( m ′ , n ′ ) with m ′ ≥ m and n ≥ n ′ .Following the lead in [6A19], we thus make the following definition. HROMATIC FIXED POINT THEORY 5
Definition 2.4.
Let r n ( G, H ) be defined so that the (
G, H, n, m ) ChromaticSmith Theorem is true if and only if m ≥ n + r n ( G, H ). Equivalently, r n ( G, H ) is the smallest r so that for all finite G –spaces X ,whenever e K ( n + r ) ∗ ( X H ) = 0 then e K ( n ) ∗ ( X G ) = 0 . The paper [BS17] shows the calculation of these ‘blue shift’ numbersis already interesting and nontrivial when (
G, H ) = ( C p , { e } ): they show r n ( C p , { e } ) = 1, from which one can deduce that r n ( G, H ) is finite in gen-eral. The paper [6A19] goes further and shows that r ≤ r n (( C p ) r , { e } ) and r n ( C p k , { e } ) ≤
1. As we will explain in §
5, these two results together implygeneral group theoretic bounds for all
H < G : r − ( G, H ) ≤ r n ( G, H ) ≤ r + ( G, H ) , where r − ( G, H ) and r + ( G, H ) are defined as follows.
Definitions 2.5.
Let H be a subgroup of a finite p –group G . (a) Let r − ( G, H ) = rank
G/H Φ( G ). Here H Φ( G ) < G is the subgroupgenerated by H and the Frattini subgroup Φ( G ), so that G/H Φ( G ) is themaximal elementary abelian p -group quotient of G with H in the kernel. (b) Let r + ( G, H ) be the minimal r such that there exists a chain ofsubgroups H = K ⊳ K ⊳ · · · ⊳ K r = G with each K i − normal in K i and K i /K i − cyclic.The lower bound for r − ( G, H ) agrees with the upper bound r + ( G, H ) insome cases, notably whenever G is abelian, when both bounds equal therank of G/H . But they don’t agree in general, as one sees already when G is the dihedral group of order 8 and H is a noncentral subgroup of order 2.There is cautious hope that r n ( G, H ) always equals the upper bound r + ( G, H ). Perhaps this hope is a bit perverse: at the end of §
5, we givesome examples showing how badly the function r + ( G, H ) behaves; e.g., it isnot always monotone in the variable H , and it is not always additive underproducts of pairs.(There is also a conjecture that the value of r n ( G, H ) is independent of n – a phenomenon seen so far in all examples, but missing any conceptualexplanation.)To possibly show that r n ( G, H ) = r + ( G, H ) in general, one needs toimprove the lower bound. By definition, a lower bound r ≤ r n ( G, H ) meansthat there exists a finite G –space X with X G not K ( n ) ∗ –acyclic, but with X H K ( n + r − ∗ –acyclic. These are hard to find in the literature; to showthat r ≤ r n (( C p ) r , { e } ), the authors of [6A19] found one family of examplesin the work of Greg Arone and Kathryn Lesh [AL20] (really going back towork of Steve Mitchell [M85]).A corollary to Theorem 2.2 offers an easier way forward. Corollary 2.6.
To show that r ≤ r n ( G, H ) , it suffices to find a finite G –space X with k n + r − ( X H ) < k n ( X G ) . KUHN AND LLOYD
As we now illustrate, explorations with very classic constructions now al-low us a much simplified proof, for all primes, of the existing lower boundfound in [6A19], and then better bounds for an infinite family of new exam-ples when p = 2.2.3. New proofs of old lower bounds at all primes. If ω is a unitaryrepresentation of a finite group G , let S ( ω ) be the sphere of unit lengthvectors. This is a G –space, and also a free S –space, where S ⊂ C actsvia scalar multiplication. The actions by G and S commute, and we let L p ( ω ) = S ( ω ) /C p , where C p < S is the group of p th roots of 1.Thus L p ( ω ) will be a lens space with a G –action. We will see that it iseasy to analyze the fixed point space L p ( ω ) G and then to compute the sizeof its Morava K –theories.We let E r denote the elementary abelian p –group C rp . Example 2.7.
Let ρ C r denote the complex regular representation of E r . Ifwe let ω = p n ρ C r , then k n + r − ( L p ( ω )) = 2 p n + r − which is less than k n ( L p ( ω ) E r ) = 2 p n + r . Details will be in § Theorem 2.8. r ≤ r n ( E r , { e } ) . As mentioned above, from this one can deduce that r − ( G, H ) ≤ r n ( G, H )for all n and H < G .2.4.
New lower bounds for the extraspecial –groups. Let D be thedihedral group of order 8, and let C < D be any one of the four noncentralsubgroups of order 2. (These are all equivalent under automorphisms of D .)Then r − ( D , C ) = 1 while r + ( D , C ) = 2, and this is the simplest examplefor which the blue shift numbers r n ( D , C ) can not be determined by theresults in [6A19].This example turns out to fit into an infinite family of examples. Let e E r denote the central product of r copies of D . This group is the extra special2–group of order 2 r associated to a quadratic form q : E r → C of Arfinvariant 0. As such, it is a nonabelian central extension C → e E r → E r .The analogue of C < D is then W r < e E r where W r is any elementaryabelian subgroup of rank r that does not contain the central C . (All suchsubgroups are equivalent.)It isn’t hard to check that r − ( e E r , W r ) = r and r + ( e E r , W r ) = r + 1 sothat r ≤ r n ( e E r , W r ) ≤ r + 1. By tweaking the construction in the previoussubsection, we show that blue shift numbers attain the upper bound. HROMATIC FIXED POINT THEORY 7
The tweak is as follows. If ω is now a real representation of a finite 2–group G , let RP ( ω ) denote the associated projective space. As before, it iseasy to analyze the fixed point space RP ( ω ) G , and then to compute the sizeof its Morava K –theories. Example 2.9.
Let e ρ r be the real regular representation of E r , pulled backto e E r . This is the sum of the 2 r distinct one dimensional real represen-tations of e E r , and e E r has one more irreductible real representation ∆ r ofdimension 2 r . If we let ω = 2 n +1 e ρ r ⊕ ∆ r , then k n + r ( RP ( ω ) W r ) = 2 n +1+2 r − r which is less than k n ( RP ( ω ) e E r ) = 2 n +1+2 r . Again invoking Corollary 2.6, this example has the following consequence.
Theorem 2.10.
For all n , r n ( e E r , W r ) = r + 1 = r + ( e E r , W r ) . A variant of this last example will prove the following.
Theorem 2.11.
For all n , r n ( e E r × E s , W r × { e } ) = r + s + 1 = r + ( e E r × E s , W r × { e } ) . This last theorem suffices to deduce the Balmer spectrum for many newgroups. We will check that it has the following consequence.
Theorem 2.12.
Let G be any 2-group fitting into a central extension C → G → E with E elementary abelian. For all K < H < G , r n ( H, K ) = r + ( H, K ) forall n . The details will be in § A new general lower bound theorem for –groups. An analysisof our argument for the extra special 2-groups leads to a general theoremthat improves the lower bound for r n ( G, H ) for many other groups too.To state this, we need a little bit of notation. Given a finite 2–group H ,let e H ∈ R [ H ] be the central idempotent e H = 1 | Φ( H ) | X h ∈ Φ( H ) h. If ω is a real representation of H , then e H ω is the maximal direct summandof ω on which Φ( H ) acts trivially, so can be viewed as a real representationof H/ Φ( H ). Theorem 2.13.
Let H be a nontrivial proper subgroup of a finite 2–group G such that Φ( H ) = Φ( G ) ∩ H , or, equivalently, H/ Φ( H ) → G/ Φ( G ) ismonic. KUHN AND LLOYD If G has an irreducible real representation ∆ such that e H Res GH (∆) is theregular real representation of H/ Φ( H ) , then, for all n , r n ( G, H ) ≥ r − ( G, H ) + 1 . The proof is in § Example 2.14. If H = W r < e E r = G , then ∆ r satisfies the hypothesis ofthe theorem. Example 2.15.
Let G be the semidirect product C ⋊ C , with C actingfaithfully on C , the group with GAP label 32 H < G be the cyclicsubgroup of order 4 which is GAP subgroup H ) = Φ( G ) ∩ H ,and G has three distinct irreducible real representations which satisfy thehypothesis of the theorem: two 2–dimensional ones that are pulled backfrom a quotient map G ։ D , and one that is faithful of dimension 4.One computes that r − ( G, H ) = 1 and r + ( G, H ) = 3. The theorem thentells us that r n ( G, H ) is either 2 or 3.2.6.
Further application of Theorem 2.2.
Our applications of Theo-rem 2.2 in the last two subsections use the theorem in its contrapositiveform: ‘if the (
G, H, n, m ) Chromatic Floyd Theorem is not true then the(
G, H, n, m ) Chromatic Smith Theorem is not true’.When combined with the upper bound r n ( C p k , { e } ) ≤
1, the direct state-ment – ‘if the (
G, H, n, m ) Chromatic Smith Theorem is true then the(
G, H, n, m ) Chromatic Floyd Theorem is true’ – implies the following the-orem.
Theorem 2.16. If C is a cyclic p –group, and X is a finite C –space, then,for all n , k n +1 ( X ) ≥ k n ( X C ) . This has interesting applications. In this paper, we illustrate this withtwo particular examples, with details in § Theorem 2.17.
Suppose C acts on the 5-dimensional Wu manifold M = SU (3) /SO (3) . Then M C will be a rational sphere. Theorem 2.18.
The real Grassmanian Gr ( R ) is K (1) –orientable, and K (1) ∗ ( Gr ( R )) is a Poincar´e duality algebra over K (1) ∗ of dimension 30. Perhaps surprisingly, our proof of this will use the Theorem 2.16 when C = C .2.7. Organization of the rest of the paper.
Section 3 has some back-ground information about the equivariant stable category and Morava K-theories needed later.In §
4, we explain how the chromatic Smith theorem problem, as stated inthe introduction, is equivalent to understanding the topology of the Balmerspectrum of the equivariant stable categories studied in [BS17]. We alsoshow how this is equivalent to understanding what chromatic type functions
HROMATIC FIXED POINT THEORY 9 can be topologically realized. Much of this material is in [BS17, 6A19], butwe hope our exposition will be of value.In §
5, we run through basic properties of the blue shift numbers r n ( G, H )and the group theoretic lower and upper bounds r − ( G, H ) and r + ( G, H ),and how two results from [6A19] are used.In § § r n ( G, H ) using representation theory. We illustrate this with the details ofExample 2.7, thus completing the proof of Theorem 2.8. We then prove ourmore delicate result, Theorem 2.13.Section 8 has the details of our results about extra special 2-groups, andhas a proof of Theorem 2.12.The details of Theorem 2.17 and Theorem 2.18 are in § G, H ) whose blueshift numbers cannot be deduced from smaller groups. We also observe thatthe particular construction we use in our new 2-group examples – RP ( ω ) –seems limited to improving the blue shift lower bound by at most 1. Theappendix has a table of exceptional pairs of 2-groups ( G, H ) for | G | ≤ Background
Background on the equivariant stable category.
In the introduc-tion, our G –spaces were not necessarily based. For comparisons with stablecategories, it is convenient to add a G -fixed disjoint basepoint to unbased G –spaces, and we generally are doing this in this paper. Theorems likeour chromatic Floyd theorems hold in either the based or unbased setting,without change.Once working in the setting of based objects, our chromatic Smith andFloyd theorems about finite G –spaces (retracts of finite G –CW complexesin the homotopy category) are easily seen to be equivalent to analogoustheorems about the compact objects in the stable homotopy category of G –spectra. We explain.The compact objects are precisely the spectra of the form S − W ∧ Σ ∞ G X with X a finite G –space and W a real representation of G . See [BGH20,Lemma 2.2]; one can also deduce this from equivariant Freudenthal theoremsas in [tD87].The stable analogue of taking H –fixed points of G –spaces is the functorthat assigns to a G –spectrum Y its geometric H –fixed point spectrum Y Φ H .This functor satisfies two basic properties: • The functor Y Y Φ H is symmetric monoidal. • There are natural symmetric monoidal equivalences(Σ ∞ G X ) Φ H ≃ Σ ∞ ( X H ) . From this, one can deduce that( S − W ∧ Σ ∞ G X ) Φ H ≃ S − W H ∧ Σ ∞ ( X H ) , and so, if Y = S − W ∧ Σ ∞ G X , then Y Φ H is K ( n ) ∗ –acyclic if and only if X H is, and, more generally, k n ( Y Φ H ) = k n ( X H ).Thus the (unstable) ( G, H, n, m ) Chromatic Smith Theorem statementis equivalent to the statement that if Y is a compact G –spectra and Y Φ H is K ( m ) ∗ –acyclic, then Y Φ G is K ( n ) ∗ –acyclic. Similarly, the ( G, H, n, m )Chromatic Floyd Theorem statement is equivalent to an evident stable ver-sion.3.2.
Background on Morava K –theories. A general reference for thissubsection is [W91].The coefficient ring of Morava K –theory is a graded field and K ( n ) is aring spectrum. These two facts imply that the natural Kunneth map × : K ( n ) ∗ ( X ) ⊗ K ( n ) ∗ K ( n ) ∗ ( Y ) → K ( n ) ∗ ( X ∧ Y )is an isomorphism for all spectra X and Y , and that the natural dualitymap K ( n ) ∗ ( Z ) → Hom K ( n ) ∗ ( K ( n ) ∗ ( Z ) , K ( n ) ∗ )is an isomorphism for all spectra Z .At all primes, K ( n ) is an associative ring spectra, and at odd primes it isalso commutative. This ensures that the functor K ( n ) ∗ ( ) : (Spectra , ∧ ) → ( K ( n ) ∗ –modules , ⊗ K ( n ) ∗ )is symmetric monoidal.There is a wrinkle when p = 2. Let t : X ∧ Y → Y ∧ X be the twistequivalence. In [W86], W¨urgler proves the formula t ∗ ( x × y ) = y × x + v n ( q ( y ) × q ( x )) , where q : K ( n ) ∗ ( X ) → K ( n ) ∗ +2 n − ( X ) is a natural derivation satisfying q = 0.From this, we can conclude that the functor X K ( n ) ∗ ( X ) is stillsymmetric monoidal if we regard K ( n ) ∗ ( X ) as taking values in Λ ∗ K ( n ) ∗ ( q )–modules, equipped with an exotic symmetric monoidal structure: M ⊗ N = M ⊗ K ( n ) ∗ N as usual, but with twist isomorphism τ : M ⊗ N → N ⊗ M given by τ ( x ⊗ y ) = y ⊗ x + v n ( q ( y ) ⊗ q ( x )).Finally we remind readers of the fundamental Thick Subcategory Theo-rem of [HS98]: If B is a proper thick subcategory of the homotopy category C of finite p –local spectra, then B = C ( n ) for some 1 ≤ n ≤ ∞ , where C ( n ) = { X | K ( n − ∗ ( X ) = 0 } , for finite n , and C ( ∞ ) = {∗} . By Ravenel’s result, we have inclusions C = C (0) ⊃ C (1) ⊃ C (2) ⊃ . . . , HROMATIC FIXED POINT THEORY 11 and we note that C ( ∞ ) = \ n< ∞ C ( n ). Each C ( n ) for n ≥ C , and, indeed, is even a prime ideal, thanks tothe Kunneth theorem.A finite spectrum X ∈ C has type n if X ∈ C ( n ) − C ( n + 1). It is anontrivial theorem of S.Mitchell [M85] that there exist type n spectra for all n and p . The idea behind an alternative proof of this by Jeff Smith will bethe basis of our proof of Theorem 2.2.4. Chromatic Smith theorems, the Balmer spectrum, and typefunctions
The homotopy category of G –equivariant spectra, with G a finite group, istensor triangulated, and in [BS17] the authors began the study of its Balmerspectrum. Among many things that they do, they are able to reduce all theirquestions to the case when G is a p group, and we will assume that this isthe case.By definition the points of the Balmer spectrum are the prime ideals in C G , the category of compact G –spectra as described above, and Balmer andSanders check that these are precisely the prime ideals P G ( H, n ) = { X ∈ C G | X Φ H ∈ C ( n ) } , with H running through representatives of the conjugacy classes of sub-groups of G , and n ≥ P G ( K, m ) ⊆ P G ( H, n ). Itisn’t hard to show that a necessary condition for this to happen is that K be subconjugate to H in G .Then one can reduce to the case when G = H : [BS17, Prop.6.11] says that P G ( K, m ) ⊆ P G ( H, n ) if and only if P H ( gKg − , m ) ⊆ P H ( H, n ) for some g ∈ G such that gKg − < H . This then connects to our earlier discussionin § Lemma 4.1.
Given
K < H , P H ( K, m + 1) ⊆ P H ( H, n + 1) if and only ifthe ( H, K, n, m ) Chromatic Smith Theorem is true.
It is useful to generalize our blue shift numbers.
Definition 4.2.
Given
K < H < G and n ≥
0, let r Gn ( H, K ) be the minimal r such that P G ( K, n + r + 1) ⊆ P G ( H, n + 1).We note that r Gn ( H, K ) is denoted i n +1 ( G ; H, K ) in [6A19].With this definition, we can restate [BS17, Prop.6.11] (assuming G is a p –group). Proposition 4.3.
Given
K < H < G , r Gn ( H, K ) = min { r n ( H, L ) | L < H is conjugate to K in G } . The connection of these problems with ‘type functions’ is emphasized in[6A19], and then generalized to the case when G is a compact Lie groupin [BGH20]. We end this section by describing how this goes, with a shortdiscussion which is perhaps a bit more direct than that in [6A19] or [BGH20].Included also is a quick proof of Proposition 4.3, plus an example illustratingthis. Lemma 4.4.
Given K (cid:12) H < G and n ≥ , there exists a finite G –space X such that X K has type m and X H has type n if and only if m ≤ n + r Gn ( H, K ) .Proof. m ≤ n + r Gn ( H, K ) if and only if P G ( K, m ) * P G ( H, n + 1), and thishappens exactly when there exists a finite G –space Y with type Y K ≥ m but type Y H ≤ n . Given such a G –space Y , let X = ( G/K + ∧ U ) ∨ ( Y ∧ V )where U has type m and V has type n , and both are given a trivial G -action.Note that ( G/K ) K = W G ( K )(= N G ( H ) /H ), which is a nonempty finite setof points, while ( G/K ) H = ∅ . Thus X K = ( W G ( K ) + ∧ U ) ∨ ( Y K ∧ V )which has type precisely m , while X H = ( ∅ + ∧ U ) ∨ ( Y H ∧ V ) = Y H ∧ V which has type precisely n . (cid:3) Let Conj( G ) denote the set of conjugacy classes of subgroups of G . Definition 4.5.
Given a finite G –space (or G –spectrum) X , its type func-tion is the function type X : Conj( G ) → N ∪ {∞} defined by type X ( H ) =type X H . Proposition 4.6.
Given a function f : Conj ( G ) → N ∪ {∞} , there existsa finite G –space X such that f = type X if and only if f ( K ) ≤ f ( H ) + r Gf ( H ) ( H, K ) for all K < H < G .Proof.
The ‘only if’ statement follows from the lemma.For the ‘if’ direction, suppose f : Conj( G ) → N ∪ {∞} satisfies f ( K ) ≤ f ( H ) + r Gf ( K ) ( H, K ) for all
K < H < G . By the lemma, for each
K < H ,there exists a finite G –space Y ( H, K ) such that type Y ( H, K ) K = f ( K )and type Y ( H, K ) H = f ( H ).For each H ∈ Conj( G ), we now let X ( H ) be the G –space defined by X ( H ) = G/H + ∧ ^ L Given K ≤ H < G , and a based finite H –space X , the based G –space G + ∧ H X satisfies:type ( G + ∧ H X ) H = type X H , and type ( G + ∧ H X ) K = min { type X L | L ≤ H and L G ∼ K } . Proof. The first statement is a special case of the second. For the second,one checks that ( G + ∧ H X ) K = _ gH ∈ ( G/H ) K X g − Kg and that gH ∈ ( G/H ) K if and only if g − Kg ≤ H . (cid:3) Proof of Proposition 4.3. Recall that our goal is to show that if K < H < G ,then r Gn ( H, K ) = min { r n ( H, L ) | L < H and L G ∼ K } . Lemma 4.4 allows usto regard this as a statement about type functions.We first check that r Gn ( H, K ) ≤ min { r n ( H, L ) | L < H and L G ∼ K } . Tosee this, let Y be a based G –space such that Y H has type n and Y K has type n + r Gn ( H, K ). Suppose L = g − Kg < H . If we consider Y as an H –spaceby restriction, then Y H has type n and Y L still has type n + r Gn ( H, K ) since Y L = g − Y K . Thus r n ( H, L ) ≥ r Gn ( H, K ).We show the other inequality holds. Given J < H with J G ∼ K , let X ( J ) be a finite H –space such that X ( J ) H has type n and X ( J ) J has type n + r n ( H, J ), and let X = ^ J X ( J ). Then X H = ^ J X ( J ) H , which still has type n , while, if L < H then X L = ^ J X ( J ) L , so that if also L G ∼ K thentype X L = max { type X ( J ) L | J G ∼ K } ≥ type X ( L ) L = n + r n ( H, L ) . Applying Lemma 4.7 to the G –space Y = G + ∧ H X , we see that Y H hastype n , whiletype Y K = min { type X L | L < H and L G ∼ K }≥ min { n + r n ( H, L ) | L < H and L G ∼ K } . This means that r Gn ( H, K ) ≥ min { r n ( H, L ) | L < H and L G ∼ K } . (cid:3) The following is likely the simplest example illustrating the differencebetween r n ( H, K ) and r Gn ( H, K ). (We thank Richard Lyons for pointing ustowards this.) Example 4.8. Let G = ( C × C ) ⋊ C , H = C × C < C × C H/K = 2, while r n ( H, L ) = rank H/L = 1. It follows that r Gn ( H, K ) = 1.5. Basic properties of r − ( G, H ) , r n ( G, H ) , and r + ( G, H ) . Here we discuss some basic properties of the blue shift numbers r n ( G, H )and their group theoretic upper and lower bounds, r + ( G, H ) and r − ( G, H ),and how these bounds are deduced from the following two results from[6A19]. Theorem 5.1. r n ( C p k , { e } ) ≤ for all n and k . This is [6A19, Thm.2.1], specialized to the case when A is cyclic. (Thoughnot noted in [6A19], the result for a general abelian group A follows fromthe cyclic group case.)We restate Theorem 2.8. Theorem 5.2. r ≤ r n ( C rp , { e } ) for all n and r . This is [6A19, Thm.2.2], and in § § H is a subgroup of a finite p –group G , r + ( G, H ) is definedto be the minimal r such that there exists a chain of subgroups H = K ⊳ K ⊳ · · · ⊳ K r = G with each K i − normal in K i and K i /K i − cyclic.We also defined r − ( G, H ) to be the rank of G/H Φ( G ). One easily seesthat r − ( G, H ) = r + ( G, H Φ( G )).The following property is elementary but very useful. Lemma 5.3. If N is normal in G , then r + ( G, H ) ≥ r + ( G/N, HN/N ) and r − ( G, H ) ≥ r − ( G/N, HN/N ) , with equality in both cases if N ≤ H . HROMATIC FIXED POINT THEORY 15 Proof. For r + , the image in G/N of a minimal subgroup chain between H and G with cyclic subquotients will be a chain between HN/N and G/N with cyclic subquotients. If N ≤ H this will be a bijection between suchchains. The statement for r − can be easily checked directly, or deduced fromthe r + case, since r − ( G, H ) = r + ( G, H Φ( H )). (cid:3) Corollary 5.4. r + ( G, H ) ≥ r − ( G, H ) .Proof. Specializing the lemma to the case when N = H Φ( G ), one learnsthat r + ( G, H ) ≥ r + ( G/H Φ( G ) , { e } ) = r + ( G, H Φ( H )) = r − ( G, H ). (cid:3) The analogue of the last lemma also holds for r n ( G, H ). Lemma 5.5. If N is normal in G , then r n ( G, H ) ≥ r n ( G/N, HN/N ) forall n , with equality if N ≤ H .Proof. Lemma 4.4 tells us there there exists a finite G/N –space X such that X G/N has type n and X HN/N has type n + r n ( G/N, HN/N ). If we regard X as a G –space via the quotient map G → G/N , then X G = X G/N hastype n and X H = X HN/N has type n + r n ( G/N, HN/N ). It follows that n + r n ( G/N, HN/N ) ≤ n + r n ( G, H ).Now suppose that N ≤ H , and that Y is a finite G –space such that Y G has type n and Y H has type n + r n ( G, H ). If we let X = Y N , then X willbe a finite G/N –space such that X G/N = Y G has type n and X HN/N = Y H has type n + r n ( G, H ). Thus n + r n ( G, H ) ≤ n + r n ( G/N, HN/N ). (cid:3) Specializing this lemma to the case when N = H Φ( G ), one learns that r n ( G, H ) ≥ r n ( G/H Φ( G ) , { e } ), which Theorem 5.2 tells us is at least a bigas the rank of G/H Φ( G ). We learn the following. Corollary 5.6. r n ( G, H ) ≥ r − ( G, H ) for all n . Another useful corollary goes as follows. Corollary 5.7. If N is normal in G , then r n ( G, N ) = r n ( G/N, { e } ) for all n . Now we note some transitivity properties. Lemma 5.8. Let H < K < G . (a) r + ( G, H ) ≤ r + ( G, K ) + r + ( K, H ) . (b) r − ( G, H ) ≤ r − ( G, K ) + r − ( K, H ) . Both of these inequalities are clear from the definitions. Strict inequalitycan certainly hold in both cases: consider { e } < C < C .An analogous property for the blue shift numbers goes as follows. Lemma 5.9. Let H < K < G . Then r n ( G, H ) ≤ r n ( G, K ) + max m ≤ n + r n ( G,K ) { r m ( K, H ) } . Proof. Let l = n + r n ( G, H ), and let X be a finite G –space such that X G hastype n and X H has type l . Let m be the type of X K . Then m ≤ n + r n ( G, K )and l ≤ m + r m ( K, H ), so that r n ( G, H ) = l − n = ( m − n ) + ( l − m ) ≤ r n ( G, K ) + r m ( K, H ) . (cid:3) Corollary 5.10. r n ( G, H ) ≤ r + ( G, H ) for all n .Proof. We prove this by induction on r + ( G, H ). r + ( G, H ) = 1 means that H is normal in G and G/H is cyclic. Butthen r n ( G, H ) = r n ( G/H, { e } ) by Corollary 5.7, and r n ( G/H, { e } ) ≤ r + ( G, H ) by Theorem 5.1.For the inductive step, let H = K ⊳ K ⊳ · · · ⊳ K r = G be a minimal chainwith each K i /K i − cyclic. By inductive hypothesis, r n ( G, K ) ≤ r + ( G, K )for all n , and r m ( K , H ) ≤ m , by the case just discussed. Thelemma applied to H < K < G then implies that r n ( G, H ) ≤ r n ( G, K ) + max m ≤ n + r n ( G,K ) { r m ( K , H ) }≤ r + ( G, K ) + 1 = r + ( G, H ) . (cid:3) Corollary 5.6 and Corollary 5.10 combine to show that r − ( G, H ) ≤ r n ( G, H ) ≤ r + ( G, H )for all H < G and all n .We end this section by noting some differences between our upper andlower bound functions. Lemma 5.11. If H < K < G , then r − ( G, H ) ≥ r − ( G, K ) . Example 5.12. Let D be the dihedral group of order 16, and let C be anoncentral subgroup of order 2. Then r + ( D , { e } ) = 2, while r + ( D , C ) =3. (We note that r − ( D , { e } ) = 2, while r − ( D , C ) = 1.) Example 5.13. Related to this last example, let D k +1 be the dihedralgroup of order 2 k +1 and let C be a noncentral subgroup of order 2. Then r − ( D k +1 , C ) = 1 while r + ( D k +1 , C ) = k . This illustrates that r + ( G, H ) − r − ( G, H ) can be arbitrarily large. Lemma 5.14. If H < G and H < G , then r − ( G × G , H × H ) = r − ( G , H ) + r − ( G , H ) . Regarding our upper bound, Lemma 5.8(a) implies that r + ( G × G , H × H ) ≤ r + ( G , H ) + r + ( G , H ) , but the next example shows that strict inequality can happen, even whenone of the pairs is as trivial as possible. HROMATIC FIXED POINT THEORY 17 Example 5.15. Let a be the generator of the cyclic group C , and let M (2)be the ‘modular maximal–cyclic group of order 16’, a group generated byelements b, c satisfying b = c = e and cbc = b . (It is group 16 C = h c i , a noncentral subgroup of order 2.Then r + ( C , { e } ) = 1 and r + ( M (2) , C ) = 2, but r + ( C × M (2) , { e } × C ) =2. To see this last fact, we have a chain of normal subgroups { e } × C = h c i ⊳ h ab , c i ⊳ h a, b, c i = C × M (2) , with h ab , c i / h c i ≃ h ab i ≃ C , and h a, b, c i / h ab , c i ≃ h a, b i / h ab i ≃ C . Remark . From this one learns that r n ( C × M (2) , { e } × C ) = 2 for all n , while r n ( M (2) , C ) can’t be determined: it is either 1 = r − ( M (2) , C ) or2 = r + ( M (2) , C ).6. Jeff Smith’s construction, and the proof of Theorem 2.2 We begin with a quick review of how idempotents in the group ring ofsymmetric groups lead to stable wedge decompositions of iterated smashproducts of spaces or spectra. The construction holds in a rather generalsetting, but we just focus on the situation that we care about.Let S ( G ) be the category of G –spectra, equipped with an associative andcommutative smash product, as in [MM02].The k th symmetric group Σ k acts naturally on the k –fold smash product X ∧ k for X ∈ S ( G ), and thus the group ring Z [Σ k ] acts naturally on X ∧ k ,viewed in the associated homotopy category ho ( S ( G )). If we fix a prime p and X is p -local, this action extends to an action by Z ( p ) [Σ k ].Recall that e ∈ Z ( p ) [Σ k ] is idempotent means that e = e , and in this case1 − e is an idempotent such that 1 = e + (1 − e ) and 0 = e (1 − e ).In this situation, one defines eX ∧ k ∈ S ( G ) to be the mapping telescope T el { X ∧ k e −→ X ∧ k e −→ X ∧ k e −→ X ∧ k e −→ . . . } . This construction satisfies various basic properties: • There is a natural equivalence X ∧ k ≃ eX ∧ k ∨ (1 − e ) X ∧ k . • There is a natural equivalence ( eX ∧ k ) Φ( H ) ≃ e ( X Φ( H ) ) ∧ k ) for all H < G . • There is a natural isomorphism E ∗ ( eX ∧ k ) ≃ eE ∗ ( X ∧ k ) for all ho-mology theories E ∗ .Now let F ∗ be a graded field of characteristic p , concentrated in evendegrees. (We will soon specialize to F ∗ = K ( n ) ∗ .) If V ∗ is a graded F ∗ –module, V ∗ will have a canonical decomposition V ∗ = V e ∗ ⊕ V o ∗ into its evenand odd graded parts. We let V ⊗ k ∗ denote the k –fold tensor product, over F ∗ ,of V ∗ with itself, viewed as a F ∗ [Σ k ]–module with the usual sign conventions.It will be convenient to let k ( d ) = (cid:0) ( p − d +1)2 (cid:1) . Proposition 6.1. Let p = 2 , and V ∗ be a finite dimensional graded F ∗ –module. For all d , there exists an idempotent e d ∈ Z (2) [Σ k ( d ) ] such that e d V ⊗ k ( d ) ∗ = 0 if and only if dim F ∗ V ∗ ≥ d. Proposition 6.2. Let p be odd, and V ∗ be a finite dimensional graded F ∗ –module. For all d , there exist idempotents e d , e ′ d ∈ Z ( p ) [Σ k ( d ) ] such that e d V ⊗ k ( d ) ∗ = 0 if and only if ( p − 1) dim F ∗ V e ∗ + dim F ∗ V o ∗ ≥ ( p − d. and e ′ d V ⊗ k ( d ) ∗ = 0 if and only if ( p − 1) dim F ∗ V o ∗ + dim F ∗ V e ∗ ≥ ( p − d. It was an insight of Jeff Smith in the mid 1980’s that the classic rep-resentation theory literature offered formulae for idempotents that wouldhave properties like those in the propositions, and that one could make re-markably good use of these, when combined with the idempotent splittingconstruction discussed earlier. See [HS98, Rav92].The idempotents e d and e ′ d are classic idempotents e λ associated to nicepartitions λ of k ( d ) = (cid:0) ( p − d +1)2 (cid:1) . For e d (for all primes), one let e d = e λ d where λ d is the maximal p –regular partition with d nonzero entries: λ d =(( p − d, ( p − d − , . . . , ( p − e ′ d = e λ ′ d ,where λ ′ d is λ d ‘transposed’, the partition having ( p − 1) entries equalling i for each d ≥ i ≥ e d satisfies the properties listed in the two propositions is given acareful proof in [Rav92, Thm.C.2.1]. Ravenel also gives references for theformulae for the e λ going back to work of H.Weyl in the 1930’s.The idempotent e ′ d is not discussed in [Rav92], but it is clear that itsatisfies e d ((Σ V ∗ ) ⊗ k ( d ) ) ≃ Σ k ( d ) e ′ d V ⊗ k ∗ , where Σ is suspension in the category of graded F ∗ –modules. As( p − 1) dim F ∗ V o ∗ + dim F ∗ V e ∗ = ( p − 1) dim F ∗ (Σ V ∗ ) e + dim F ∗ (Σ V ∗ ) o , the second statement of Proposition 6.2 follows from the first.Now we consider the implication of the propositions for the spectra e d X ∧ k ( d ) ,and, when p is odd, e ′ d X ∧ k ( d ) . Corollary 6.3. Let p be odd, and X be a finite spectrum. (a) e d X ∧ k ( d ) has type ≤ n if and only if ( p − 1) dim K ( n ) ∗ K ( n ) e ∗ ( X ) + dim K ( n ) ∗ K ( n ) ∗ ( X ) o ∗ ≥ ( p − d. (b) e ′ d X ∧ k ( d ) has type ≤ n if and only if ( p − 1) dim K ( n ) ∗ K ( n ) o ∗ ( X ) + dim K ( n ) ∗ K ( n ) ∗ ( X ) e ∗ ≥ ( p − d. HROMATIC FIXED POINT THEORY 19 Proof. This follows immediately from Proposition 6.2, as the Kunneth iso-morphism for K ( n ) ∗ tells us that K ( n ) ∗ ( e d X ∧ k ( d ) ) = e d K ( n ) ∗ ( X ∧ k ( d ) ) = e d K ( n ) ∗ ( X ) ⊗ k ( d ) , with a similar description for K ( n ) ∗ ( e ′ d X ∧ k ( d ) ). (cid:3) Corollary 6.4. Let p = 2 , and X be a finite spectrum. Then e d X ∧ k ( d ) hastype ≤ n if and only if dim K ( n ) ∗ K ( n ) ∗ ( X ) ≥ d .Proof. We need a little bit of extra care to implement the proof as in theodd prime corollary, because of the extra wrinkle with the Kunneth theoremfor Morava K –theory at the prime 2.Recall that the assignment X K ( n ) ∗ ( X ) is only symmetric monoidal ifwe view K ( n ) ∗ ( X ) as a Λ ∗ K ( n ) ∗ ( q )–module with the twist isomorphism givenby τ ( x × y ) = y ⊗ x + v n qy ⊗ qx .Let ¯ ⊗ be this exotic tensor product, and ⊗ the standard one. We claimthat, if e ∈ K ( n ) ∗ [Σ k ] is any idempotent and M is any graded Λ ∗ K ( n ) ∗ ( q )–module, then dim K ( n ) ∗ eM ¯ ⊗ k = dim K ( n ) ∗ eM ⊗ k . Assuming this, the corol-lary follows as in the odd prime case, as K ( n ) ∗ ( e d X ∧ k ( d ) ) = e d K ( n ) ∗ ( X ) ¯ ⊗ k ( d ) . To check this claim, we filter M as a Λ ∗ K ( n ) ∗ ( q )–module in the simplestway possible: let F M = ker q , and then let F M = M . This induces afiltration on M ¯ ⊗ k as a K ( n ) ∗ [Σ k ]–module, with associated graded module( F M ⊕ M/F M ) ¯ ⊗ k . But this is just ( F M ⊕ M/F M ) ⊗ k , since q actstrivially on ( F ⊕ M/F ). Since idempotents commute with filtrations, wehavedim K ( n ) ∗ eM ¯ ⊗ k = dim K ( n ) ∗ e ( F M ⊕ M/F M ) ¯ ⊗ k = dim K ( n ) ∗ e ( F M ⊕ M/F M ) ⊗ k = dim K ( n ) ∗ eM ⊗ k . (cid:3) We will use the following elementary lemma in our proof of Theorem 2.2. Lemma 6.5. Let p be an odd prime. Suppose that nonnegative integers a e , a o , b e , and b o satisfy a e + a o < b e + b o . Then there exists d such that at leastone of the following inequalities holds: ( p − a e + a o < ( p − d ≤ ( p − b e + b o , ( p − a o + a e < ( p − d ≤ ( p − b o + b e . Proof. The inequality a e + a o < b e + b o implies that ( b e − a e ) + ( b o − a o ) ≥ b e − a e ) ≥ b o − a o ) ≥ b e − a e ) ≥ 1, then ( p − b e − a e ) ≥ p − 2. Adding that to ( b e − a e ) +( b o − a o ) ≥ p − b e + b o ] − [( p − a e + a o ] ≥ p − 1. This, inturn, clearly implies that there exists d such that( p − a e + a o < ( p − d ≤ ( p − b e + b o . The case when ( b o − a o ) ≥ (cid:3) Proof of Theorem 2.2. Suppose that a finite G –spectrum X satisfiesdim K ( m ) ∗ K ( m ) ∗ ( X Φ( H ) ) < dim K ( n ) ∗ K ( n ) ∗ ( X Φ( G ) ) . We need to show that there is then a finite G –spectrum F with K ( m ) ∗ ( F Φ( H ) ) =0 and K ( n ) ∗ ( F Φ( G ) ) = 0.If p = 2, we let d = dim K ( n ) ∗ K ( n ) ∗ ( X Φ( G ) ), and then let F = e d X ∧ k ( d ) .By Corollary 6.4, F has the desired property.If p is odd, the lemma shows that there exists a d such that at least oneof the following is true: (a) ( p − 1) dim K ( m ) ∗ K ( m ) e ∗ ( X Φ( H ) ) + dim K ( m ) ∗ K ( m ) o ∗ ( X Φ( H ) ) < ( p − d ≤ ( p − 1) dim K ( n ) ∗ K ( n ) e ∗ ( X Φ( G ) ) + dim K ( n ) ∗ K ( n ) o ∗ ( X Φ( G ) ) , (b) ( p − 1) dim K ( m ) ∗ K ( m ) o ∗ ( X Φ( H ) ) + dim K ( m ) ∗ K ( m ) e ∗ ( X Φ( H ) ) < ( p − d ≤ ( p − 1) dim K ( n ) ∗ K ( n ) o ∗ ( X Φ( G ) ) + dim K ( n ) ∗ K ( n ) e ∗ ( X Φ( G ) ) . If (a) holds, let F = e d X ∧ k ( d ) . If (b) holds, let F = e ′ d X ∧ k ( d ) . By Corol-lary 6.3, F does the job. (cid:3) Lower bounds for r n ( G, H ) using representation theory In this section, we give the background needed to use our lens space andprojective space constructions, and give the details of Example 2.7 (whichimplies that r ≤ r n ( C rp , { e } )), and then our more delicate Theorem 2.13.7.1. The fixed points of L p ( ω ) and RP ( ω ) . As in the introduction, if ω is a unitary representation of a finite group G , we let L p ( ω ) = S ( ω ) /C p ,where S ( ω ) is the unit sphere in ω , and C p ⊂ U (1) ⊂ C × is the group of p throots of unity. Thus, if ω has complex dimension d , then L p ( ω ) = L p ( C d ) isa (2 d − G .To describe L p ( ω ) G , we need to recall that ω admits a canonical de-composition into its isotypical components: ω ≃ M i ω i , with the sumrunning over an indexing set for the simple C [ G ]–modules λ i . Explicitly, ω i = Hom C [ G ] ( λ i , ω ) ⊗ C λ i . In particular if λ i is a one dimensional complexrepresentation, then dim C ω i = dim C Hom C [ G ] ( λ i , ω ), and this equals thenumber of copies of λ i in ω . Lemma 7.1. Given w ∈ S ( ω ) , [ w ] ∈ L p ( ω ) is fixed by G if and onlyif w spans a 1–dimensional sub-representation of ω which factors through G/ Φ( G ) .Proof. Given w ∈ S ( ω ), [ w ] ∈ L p ( ω ) is fixed by G if and only if we candefine a character λ : G → C p ⊂ C × by gw = λ ( g ) w , and such a characterwill factor through G/ Φ( G ). (cid:3) HROMATIC FIXED POINT THEORY 21 The lemma has the following formula as a corollary. Proposition 7.2. L p ( ω ) G = a i L p ( ω i ) , with the disjoint union runningover the 1–dimensional complex representations of G which factor through G/ Φ( G ) . Similarly, if ω is a real representation of a finite group G , we let RP ( ω ) bethe associated projective space: the G –space of real lines in ω . So if ω is d –dimensional, then RP ( ω ) = RP ( R d ) is a ( d − G .The isotypical decomposition again takes the form ω ≃ M i ω i , with thesum running over an indexing set for the simple R [ G ]–modules λ i , but now ω i = Hom R [ G ] ( λ i , ω ) ⊗ F i λ i where F i is the field End R [ G ] ( λ i ) (either R , C , or H ). In particular, if λ i is a one dimensional real representation then F i = R ,so ω i is a real vector space of dimension equal to dim R Hom R [ G ] ( λ i , ω ), whichequals the number of copies of λ i in ω .Just as before, the fixed point space of RP ( ω ) is easily computed. Proposition 7.3. RP ( ω ) G = a i RP ( ω i ) , with the disjoint union runningover the 1–dimensional real representations of G . As in § G , we let e G ∈ R [ G ] be the centralidempotent e G = 1 | Φ( G ) | X g ∈ Φ( G ) g. If ω is a real representation of G , then e G ω is the maximal direct summandof ω on which Φ( G ) acts trivially, so can be viewed as a real representationof G/ Φ( G ) pulled back to G . The representation e G ω is isomorphic to e H ω ≃ M i ω i , with the sum over the 1–dimensional real representations of G , so the proposition has the following corollary. Corollary 7.4. The summand inclusion e G ω ֒ → ω induces a homeomor-phism RP ( e G ω ) G = RP ( ω ) G . The Morava K –theory of L p ( C d ) and RP ( R d ) . We will use thefollowing well known calculations. Proposition 7.5. When p = 2 , one has k n ( RP ( R d )) = d if d ≤ n +1 n +1 if d is even and d ≥ n +1 n +1 − if d is odd and d > n +1 . Proposition 7.6. For all primes p , one has k n ( L p ( C d )) = ( d if d ≤ p n p n if d ≥ p n . We sketch the proofs.In both cases, we compute using the Atiyah–Hirzebruch spectral sequenceconverging to K ( n ) ∗ ( X ). This has E ∗ , ∗ = H ∗ ( X ; Z /p )[ v ± n ], and first possi-ble nonzero differential given by d p n − ( x ) = v n Q n ( x ), where Q n is the n thMilnor primitive in the mod p Steenrod algebra.For the first proposition, one knows that H ∗ ( RP ( R d ); Z / 2) = Z / x ] / ( x d ).From the definition of Q n , one sees that Q n ( x ) = x n +1 if 2 n +1 < d . As Q n is a derivation, it follows that Q n ( x r ) = x n +1 + r if r is odd and 2 n +1 + r < d ,and is 0 otherwise. It follows that nonzero elements in the E n +1 –term of thespectral sequence consists of the even dimensional classes between degrees0 and 2 n +1 , and the odd dimensional classes between degrees d + 1 − n +1 and d . Even dimensional classes are in the image of K ( n ) ∗ ( CP ∞ ) under thecomposite RP ( R d ) ֒ → RP ∞ → CP ∞ , and so are permanent cycles. It followsthat there can be no higher differentials, and the proposition follows.When p = 2, the first proposition includes the second.The proof of the second proposition when p is odd is similar, starting fromthe calculations H ∗ ( L p ( C d ); Z /p ) = Λ ∗ ( x ) ⊗ Z /p [ y ] / ( y d ), with Q n ( x ) = y p n if p n < d .7.3. Using L p ( ω ) : the details of Example 2.7. Recall the situation ofExample 2.7. Let E r = ( C p ) r and let ρ C r denote its regular representation:the sum of the p r distinct 1–dimensional complex representations of E r .We let ω = p n ρ C r , and we want to show that k n + r − ( L p ( ω )) = 2 p n + r − and k n ( L p ( ω ) E r ) = 2 p n + r . As p n ρ C r is p n + r dimensional, L p ( ω ) = L p ( C p n + r ), and the first followsimmediately from Proposition 7.6.As ω is the direct sum of p n copies of each of the p r E r , Proposition 7.2 tells us that L p ( ω ) E r is the disjoint unionof p r copies of L p ( C p n ), and the second calculation also follows from Propo-sition 7.6.7.4. Proof of Theorem 2.13. We recall the hypotheses of Theorem 2.13.We are assuming that H be a proper nontrivial subgroup of a finite 2–group G such that Φ( H ) = Φ( G ) ∩ H , and that G has an irreducible realrepresentation ∆ such that e H Res GH (∆) is the regular real representation of H/ Φ( H ) pulled back to H .We wish to show that then, for all n , r n ( G, H ) ≥ r − ( G, H ) + 1. HROMATIC FIXED POINT THEORY 23 Let a = the rank of G/ Φ( G ) and let b = the rank of H/ Φ( H ). Thehypothesis Φ( H ) = Φ( G ) ∩ H means that H/ Φ( H ) → G/ Φ( G ) is monic,and thus r − ( G, H ) = a − b . So we wish to show that r n ( G, H ) ≥ a − b + 1.Let e ρ G be the regular representation of G/ Φ( G ) pulled back to G , and,similarly, let e ρ H be the regular representation of H/ Φ( H ) pulled back to H .Fixing n , let ω = 2 n +1 e ρ G ⊕ ∆ . By Corollary 2.6, r n ( G, H ) ≥ a − b + 1 will follow if we can show that k n + a − b ( RP ( ω ) H ) < k n ( RP ( ω ) G ) . The next two lemmas say what we need. Lemma 7.7. k n ( RP ( ω ) G ) = 2 n + a +1 Lemma 7.8. k n + a − b ( RP ( ω ) H ) = 2 n + a +1 − b .Proof of Lemma 7.7. Each of the 2 a G occurs exactly 2 n +1 times in ω , and thus the corresponding isotypicalcomponents of ω all have dimension 2 n +1 . Thus k n ( RP ( ω ) G ) = 2 a k n ( RP ( R n +1 )) = 2 a · n +1 = 2 n + a +1 . (cid:3) Proof of Lemma 7.8. By Corollary 7.4, RP ( ω ) H = RP ( e H ω ) H . We analyze e H Res GH ( ω ).Since H/ Φ( H ) has index 2 a − b in G/ Φ( G ), we have that Res GH ( e ρ G ) =2 a − b e ρ H , and e H acts as the identity on this. Meanwhile, we have assumedthat e H Res GH (∆) = e ρ H . Thus e H Res GH ( ω ) = e H Res GH (2 n +1 e ρ G ⊕ ∆)= (2 n +1 a − b + 1) e ρ H = (2 n + a − b +1 + 1) e ρ H . This implies that each of the 2 b H occurs2 n + a − b +1 + 1 times in ω , and thus the corresponding isotypical componentsof ω all have dimension 2 n + a − b +1 + 1. Thus k n + a − b ( RP ( ω ) H ) = 2 b k n ( RP ( R n + a − b +1 +1 )) = 2 b · (2 n + a − b +1 − 1) = 2 n + a +1 − b . (cid:3) Example 7.9. It is worth seeing explicitly how and why this all works in thesimplest example, when n = 0, G = D and H = C , a noncentral subgroupof order 2. D has four 1-dimensional representations λ , . . . , λ , and one 2–dimensionalirreducible ∆, which, when restricted to C is 1 ⊕ σ , the sum of the two 1-dimensional representations of C .We let ω = 2( λ ⊕ λ ⊕ λ ⊕ λ ) ⊕ ∆ , a 10–dimensional representation of D . Thus RP ( ω ) is the space RP ( R ) = RP with an action of D , and we have( RP ) D = a RP ( R ) = a RP , So that k ( RP ) D = 4 · H , ω = 5(1 ⊕ σ ) . Thus ( RP ) C = a RP ( R ) = a RP . Since Q acts nontrivially on H ∗ ( RP ; Z / 2) (not true with RP replaced by RP !), we have that k ( RP ) = 3, so that k ( RP ) C = 2 · Blue shift numbers for extraspecial 2-groups Extraspecial 2–groups and their real representations. We col-lect some information we will need about extraspecial 2–groups and theirreal representations. A general reference for the group theory is [Asch00,Chapter 8], and [Q71] has what we need about the representation theory.By definition, an extraspecial 2–group is a finite 2–group e E such that e E ′ = Φ( e E ) = Z ( e E ) is cyclic of order 2. Thus it is a nonabelian group thatfits into a central extension C i −→ e E π −→ E, with E elementary abelian.One defines q : E → C by the formula q ( a ) = c if π (˜ a ) = a and ˜ a = i ( c ),and then h , i : E × E → C by h a, b i = q ( a + b ) − q ( a ) − q ( b ). Then h , i is nondegenerate, symmetric, and bilinear, and q is a quadratic form. Thesethen determine the group structure on e E by the formulae ˜ a = i ( q ( π (˜ a )))and [˜ a, ˜ b ] = i ( h π (˜ a ) , π (˜ b ) i ).Quadratic forms like this are classified by their Arf invariant: one learnsthat E must be of even dimension, and that, up to isomorphism, there aretwo distinct possible quadratic functions q on E r . The one that will concernus has Arf invariant 0: q = x y + · · · + x r y r , where ( x , . . . , x r , y , . . . , y r )is dual to a basis ( a , . . . , a r , b , . . . , b r ) for E .It follows that e E r has generators ˜ a , . . . , ˜ a r , ˜ b , . . . , ˜ b r , c with c = e , ˜ a i =˜ b i = c for all i , and with all generators commuting except that [˜ a i , ˜ b i ] = c for all i .A subspace W < E r is isotropic if h W, W i = 0. It is not hard to seethat, under π : e E r → E r , maximal elementary abelian subgroups of e E r not containing the center h c i will correspond to maximal isotropic subspacesof E r , and all such will be equivalent to W r , the subgroup generated by˜ a , . . . , ˜ a r . HROMATIC FIXED POINT THEORY 25 Our group e E r is sometimes denoted 2 r + in the literature, and can alsobe described as D ◦ r , the central product of r copies of the dihedral group D of order 8. The other extra special 2–group of order 2 r , sometimesdenoted 2 r − , is Q ◦ D ◦ r − , where Q is the quaternionic group of order 8.The 2 r E r pullback to give 1–dimensional real representations of e E r . The group e E r has one more ir-reducible real representation ∆ r , a faithful representation of dimension 2 r on which c acts as − 1. Of key importance to us is that ∆ r restricted to W r is the regular representation of W r [Q71, (5.1)].8.2. The computation of r n ( e E r , W r ) and r n ( e E r × E s , W r × { e } ) . Wecompute r n ( e E r , W r ). We have thatΦ( W r ) = { e } = Φ( e E r ) ∩ W r , and ∆ r restricted to W r is the regular representation, so the hypothesesof Theorem 2.13 hold. As r − ( e E r , W r ) = r , we deduce the conclusion ofTheorem 2.10: r n ( e E r , W r ) = r + 1 = r + ( e E r , W r ).To fill in the details of Example 2.9, we let ω = 2 n +1 e ρ r ⊕ ∆ r , where e ρ r is the real regular representation of E r , pulled back to e E r .Lemma 7.7 tells us that k n ( RP ( ω ) e E r ) = 2 n +1+2 r , while Lemma 7.8 tells us that k n + r ( RP ( ω ) W r ) = 2 n +1+2 r − r . Similarly, Theorem 2.11 is the special case of Theorem 2.13, applied to thepair ( e E r × E s , W r × { e } ), with the special representation of e E r × E s chosento be ∆ r , pulled back to the product. Now r − ( e E r × E s , W r × { e } = r + s ,so we learn that r n ( e E r × E s , W r × { e } ) = r + s + 1 = r + ( e E r × E s , W r × { e } ) . Blue shift numbers for a family of groups. Let G be the collectionof 2–groups G fitting into a central extension C → G p −→ E with E elementary abelian. The topology of the Balmer spectrum for any G ∈ G is determined by Theorem 2.12 which we restate. Theorem 8.1. Let G ∈ G . For all K < H < G , r n ( H, K ) = r + ( H, K ) forall n . Here we show that the family of calculations r n ( e E r × E s , W r × { e } ) = r + s + 1 = r + ( e E r × E s , W r × { e } )suffices to prove this.We start with some elementary observations. Note that if G is in G and H < G is nontrivial, then H is again in G . Thusit suffices to show that if G ∈ G then r n ( G, H ) = r + ( G, H ) for all H < G .We prove this by induction on | G | .For any group G ∈ G , either G is abelian or G ′ = Φ( G ) = C . If G isabelian we are done: r n ( G, H ) = r + ( G, H ) for all H < G . Thus we canassume this is not the case.Next observe that if G ∈ G and N ⊳G is any proper normal subgroup, then G/N ∈ G . Now let N = H ∩ Z ( G ) which will be a normal subgroup of G .Then r n ( G, H ) = r n ( G/N, H/N ) and r + ( G, H ) = r + ( G/N, H/N ). If N = { e } then r n ( G/N, H/N ) = r + ( G/N, H/N ) by our inductive assumption,and we are done. Thus we can assume that H ∩ Z ( G ) = { e } .Since Φ( G ) ≤ Z ( G ), we see that H ∩ Φ( G ) = { e } also. This implies that p : H → E is monic, so H is elementary abelian.We isolate the next part of our argument as a lemma. Lemma 8.2. In this situation, suppose that C G ( H ) contains an element oforder 4. Then r − ( G, H ) = r + ( G, H ) , and so r n ( G, H ) = r + ( G, H ) .Proof. Suppose that there exists x ∈ C G ( H ) of order 4. As H is elementaryabelian, we know that x H . Since x must generate Φ( G ), we can furtherconclude that x H Φ( G ), which means that p ( x ) p ( H ). If we let K < G be the subgroup generated by H and x , then we have r + ( G, H ) ≤ r + ( G, K )= 1 + r + ( E, p ( K ))= 1 + r − ( E, p ( K ))= r − ( E, p ( H ))= r − ( G, H ) , (cid:3) Since Z ( G ) < C G ( H ), the lemma implies that we can assume that Z ( G )is elementary abelian, and thus admits a decomposition Z ( G ) = C × E s forsome s , where the first factor is G ′ . If we let e E = G/E s then e E will be anextraspecial 2–group, so e E/ e E ′ = E r , for some r , and the sequence C → G p −→ E identifies with a sequence of the form C → e E × E s p × −−→ E r × E s . Now recall that H < G = e E × E s is elementary abelian and that H ∩ Z ( G )is trivial. Since Z ( G ) = C × E s , we conclude that H projects isomorphicallyto an elementary abelian subgroup in e E that does not contain the central C .Another was of putting this, is that H is the graph of a homomorphism W → E s , where W < ˜ E is an elementary abelian subgroup that does not containthe central C . One can conclude that C G ( H ) = C G ( W ) = C e E ( W ) × E s . HROMATIC FIXED POINT THEORY 27 If e E = Q ◦ D ◦ r − , then Q < C e E ( W ), and so the centralizer contains anelement of order 4, and the lemma applies. Similarly, there is an element oforder 4 in C e E ( W ) if e E = e E r and W has rank less than r .So we can assume that our pair ( G, H ) = ( e E r × E s , H ) where H is thegraph of a homomorphism W r → E s . But it is easy to check that this pair isequivalent to ( e E r × E s , W r ×{ e } ), so r n ( G, H ) = r + ( G, H ) by Theorem 2.11.9. Further applications Here we give the details of our applications of the statement ‘if the( G, H, n, m ) Chromatic Smith Theorem is true then the ( G, H, n, m ) Chro-matic Floyd Theorem is true’.At the moment, all our knowledge of when Chromatic Smith theoremsare true are consequences of the calculation r n ( C p k , { e } ) ≤ 1, and the cor-responding chromatic Floyd theorem is as we stated in Theorem 2.16: if C is a cyclic p –group and X is a finite C –space, then, for all n ,dim K ( n +1) ∗ K ( n + 1) ∗ ( X ) ≥ dim K ( n ) ∗ K ( n ) ∗ ( X C ) . As our first application, we note that this implies that one has chromaticanalogues of the classical theorem that if a p –group acts on a mod p homol-ogy sphere, its fixed point space is again a mod p –homology sphere.We illustrate this with Theorem 2.17 which we repeat here. Theorem 9.1. Suppose C acts on the 5-dimensional Wu manifold M = SU (3) /SO (3) . Then M C will be a rational sphere.Proof. H ∗ ( M ; Z / 2) has a basis given by classes x , x , x , x in degrees0,2,3,5 such that Sq ( x ) = x and Sq ( x ) = x . From this, one seesthat Q ( x ) = x , and then that dim K (1) ∗ K (1) ∗ ( M ) = 2. Since the( C , { e } , , 1) Chromatic Smith Theorem is true, Theorem 2.2 tells us thatdim Q H ∗ ( M C ; Q )) is at most 2.The possibility that dim Q H ∗ ( M C ; Q )) = 0, i.e. that M C = ∅ , is ruledout because M is not a boundary, and a standard exercise then shows that M can’t admit a free action of C . The possibility that dim Q H ∗ ( M C ; Q )) =1 is ruled out by Euler characteristic considerations as χ ( M C ) ≡ χ ( M )mod 2 [F52, Thm.4.2]. Thus dim Q H ∗ ( M C ; Q )) = 2, and so M C is arational sphere. (cid:3) Our second application of the ‘positive’ direction of Theorem 2.2 con-cerns computing K ( n ) ∗ ( X ) for n ≥ K ( n ) ∗ ( X ) is d p n − with formula d p n − ( x ) = v n Q n x for x ∈ H ∗ ( X ; K ( n ) ∗ ) = H ∗ ( X ; Z /p )[ v ± n ]. It follows that, if we let H ( X ; Q n ) = ker { Q n : H ∗ ( X ; Z /p ) → H ∗ ( X ; Z /p ) } im { Q n : H ∗ ( X ; Z /p ) → H ∗ ( X ; Z /p ) } , and then let k Q n ( X ) = dim Z /p H ( X ; Q n ), then k Q n ( X ) will equal the di-mension of the 2 p n th page of the AHSS as a K ( n ) ∗ –vector space. Thus weget the upper bound k n ( X ) ≤ k Q n ( X ).Meanwhile, one has the lower bound k n − ( X C ) ≤ k n ( X ) if X admits anaction of a cyclic p –group C .If the lower bound matches the upper bound, we get the conclusion of thenext theorem. Theorem 9.2. Suppose a finite complex X admits an action of a cyclic p –group C , such that k n − ( X C ) = k Q n ( X ) . Then the AHSS computing K ( n ) ∗ ( X ) collapses at E ∗ , ∗ p n and k n ( X ) = k Q n ( X ) . We illustrate this by sketching the proof of Theorem 2.18: a calculationof K (1) ∗ ( Gr ( R )), the mod 2 K -theory of the Grassmanian of 2–planes in R , a 100 dimensional oriented manifold.A calculation (first done with a computer) of the Q –Margolis homol-ogy group of the 1326 dimensional algebra H ∗ ( Gr ( R ); Z / 2) shows that H ( Gr ( R ); Q ) is 30 dimensional, and, indeed, is still a Poincar´e dualityalgebra with top class in degree 100. Most of the Margolis homology is ineven degrees, but not all, and there is still room for higher differential in theAHSS.Now we bring in equivariance. Let 1, σ , and τ be the irreducible realrepresentations of C : 1 is the trivial representation, σ is the sign represen-tation, and τ is the restriction of a 1–dimensional complex representation.Now let ω = 2(1 ⊕ σ ) ⊕ τ , a 52 dimensional real representation of C .Similar to Proposition 7.3, one can analyze the fixed points of Gr ( ω ) andone finds that Gr ( ω ) C = Gr ( R ) × Gr ( R ) ∐ Gr ( R ) ∐ Gr ( R ) ∐ Gr C ( C )= S × S ∐ {∗} ∐ {∗} ∐ C P , so that k ( Gr ( ω ) C ) = 4 + 1 + 1 + 24 = 30.Since 30=30 (!), we conclude that the AHSS collapses at E ∗ , ∗ , and so Gr ( R ) is K (1)–orientable and K (1) ∗ ( Gr ( R )) is a Poincar´e duality al-gebra over K (1) ∗ of dimension 30. Remark . In work in progress, we have similarly calculated K ( n ) ∗ ( Gr d ( R m ))for all n and m when d = 2, and believe our methods will work for gen-eral d . The difficult part of the work is the calculation of k Q n ( Gr d ( R m )),which miraculously seems to always agree with the much easier calculationof k n − ( Gr d ( ω ) C ) for a well chosen C –representation ω .10. Essential pairs and final remarks Essential pairs. If one wishes to systematically try to prove that r n ( G, H ) = r + ( G, H ) for all H < G , one can focus on potential minimalcounterexamples. By pulling back actions through quotient maps, thesemust be pairs as in the following definition. HROMATIC FIXED POINT THEORY 29 Definition 10.1. If H is a subgroup of a finite p –group G , say ( G, H ) isan essential pair , if r + ( G, H ) > r + ( G/N, HN/N ) for all nontrivial normalsubgroups N ⊳ G .Clearly if ( G, H ) is essential, it is necessary that H contain no nontrivialnormal subgroups of G , so, in particular, H ∩ Z ( G ) = { e } . Also, since r − ( G, H ) = r + ( G/ Φ( G ) , H Φ( G ) / Φ( G )), we see that if ( G, H ) is essential,then either r − ( G, H ) < r + ( G, H ) or Φ( G ) is trivial (i.e., G is elementaryabelian).To easily identify essential pairs, the following lemma is useful. Lemma 10.2. ( G, H ) is essential if r + ( G, H ) > r + ( G/C, HC/C ) for allcentral subgroups C < G of order p .Proof. We prove the lemma in its contrapositive form.The group G acts on any normal subgroup N by congugation, and thefixed point set identifies with N ∩ Z ( G ). As G is a p –group, the numberof fixed points must be congruent to 0 mod p . As e ∈ N is clearly fixed,we conclude that N ∩ Z ( G ) is nontrivial, and thus N contains a centralsubgroup C of order p .Since r + ( G, H ) ≥ r + ( G/C, HC/C ) ≥ r + ( G/N, HN/N ) holds in general,if r + ( G, H ) = r + ( G/N, HN/N ) then r + ( G, H ) = r + ( G/C, HC/C ). (cid:3) Examples 10.3. The only essential pairs ( G, H ) with G abelian are thepairs ( E r , { e } ), for r ≥ Examples 10.4. At the prime 2, the pairs ( e E r × E s , W r ×{ e } ) are essential,and, up to equivalence, there are no other essential pairs ( G, H ) with G inthe family of groups for which Theorem 2.12 applies. Examples 10.5. If p is odd, let e E r denote the extraspecial group of order p r having exponent p . Just as in the p = 2 case, we let W r denote anyelementary abelian p –subgroup of rank r that doesn’t contain the center.Then ( e E r × E s , { e } ) is essential, with r − ( e E r × E s , { e } ) = 2 r + s < r + s = r + ( e E r × E s , { e } ) , as are the pairs ( e E r × E s , W r × { e } ), with r − ( e E r × E s , W r × { e } ) = r + s < r + s = r + ( e E r × E s , W r × { e } ) . In the appendix, we include tables of all essential pairs ( G, H ) with G a nonabelian 2–group of order up to 32. Here we highlight a few of thesethat we feel are the pairs that need to be understood if any more significantprogress is to be made on the Chromatic Smith Theorem problem. Example 10.6. Let G be the group with GAP label 16 C ⋊ C , with C acting on C via the quotient C ։ C .It also fits into a central extension C → G → C × C , so the group is ‘almost’ in our family of friendly groups dealt with in Theo-rem 2.12, but not quite.Then ( G, { e } ) is essential, r − ( G, { e } ) = 2 and r + ( G, { e } ) = 3. Example 10.7. Let M (4) be the group with GAP label 16 C → M (4) → C × C , and also a central extension C → M (4) → C × C , so again the group is almost, but not quite, in our family of friendly groups.Let C < M (4) be any of the noncentral subgroups of order 2, e.g.GAP subgroup M (4) , C ) is essential, r − ( M (4) , C ) = 1 and r + ( M (4) , C ) = 2.It is interesting to note that, though ( M (4) , C ) is essential, ( M (4) × C , C × { e } ) is not, as the calculation in Example 5.15 shows. Example 10.8. The dihedral group D has GAP label 16 Z ( D ) = C < C = D ′ = Φ( D ), so D can be written as a noncentralextension C → D → C × C . Let C < D be any of the noncentral subgroups of order 2, e.g. GAPsubgroup D , C ) is essential, r − ( D , C ) = 1 and r + ( D , C ) =3. Our last example illustrates how complicated things becomes as one ex-amines groups of order 32. Example 10.9. Let G be the group with GAP label 32 C ⋊ C , with C acting faithfully on C .Then ( G, H ) is essential when H is any of the inequivalent subgroups withGAP number The limitations of our RP ( ω ) and L p ( ω ) constructions. It isworth pondering why we are able to prove interesting lower bound theo-rems using the RP ( ω ) and L p ( ω ) constructions, and how these theorems arelimited.Informally, the fixed point formulae for RP ( ω ) G shows that there is amod 2 cohomology class for each 1–dimensional real representation in ω ,and these are arranged in ‘piles’ corresponding to the distinct representa-tions. When one considers RP ( ω ) H , these piles get ‘stacked up’ when distinctrepresentations becomes the same when restricted to H , and there are newclasses coming from higher dimensional irreducible summands of ω that have1–dimensional summands when restricted to H .In the proof of the elementary abelian lower bound, Theorem 2.8, the ac-tion of the Milnor Q m ’s on the piles align just right to show that r − ( G, H ) ≤ r n ( G, H ). To do better, we need k m ( RP ( ω ) H ) to be made smaller, and this HROMATIC FIXED POINT THEORY 31 means that we need some new 1-dimensional H –representations to cancelsome of those pulled back from G , via the operation Q m . Since, as a functionof d , k m ( RP ( R d )) goes up and down only by 1 once d is large, we see that | H/ Φ( H ) | , the number of 1–dimensional representation of H , is the mostthat we can lower k m ( RP ( ω ) H ), by adding higher dimensional irreducible G –representations to ω . When one ponders the numbers, it becomes clear thatthis is not enough of a change to prove more than r − ( G, H ) + 1 ≤ r n ( G, H ).In the odd prime situation, the situation is even worse: a 1-dimensionalcomplex representation of H contributes both an odd and an even dimen-sional class to the mod p cohomology of L p ( ω ) H , and if we add such a pairlike this coming from a new 1–dimensional H –representation, Q m may pairan old odd class with the new even class, but the new odd class will stillbe left. Otherwise said, k m ( L p ( C d )) is constant once d is large, and we cannever use this method to prove more than r − ( G, H ) ≤ r n ( G, H ). Appendix A. Essential pairs ( G, H ) with | G | ≤ Essential Pairs for Nonabelian Groups of Order 8 and 16 G [GAP label] H [GAP subgroup] r − ( G, H ) r + ( G, H ) r n ( G, H ) D [8, 3] C [3] 1 2 2 C ⋊ C [16, 3] { e } [1] 2 3 M (2) [16, 6] C [3] 1 2 D [16, 7] C [3] 1 3 ≥ SD [16, 8] C [3] 1 3 ≥ C × D [16, 11] C [3] 2 3 3 Essential Pairs for Nonabelian Groups of Order 32 G [GAP label] H [GAP subgroup] r − ( G, H ) r + ( G, H ) r n ( G, H ) C ⋊ C [32, 6] C [3], C [14], C [19] 2, 2, 1 3, 3, 2 C [9], C [24] 1, 1 3, 3 ≥ ≥ C .D [32, 7] C [3] 1 3 ≥ C [7], C [20] 2, 1 3, 2 C ≀ C [32, 11] C [3] 1 3 ≥ C [7] 2 3 M (2) [32, 17] C [3] 1 2 D [32, 18] C [3] 1 4 ≥ SD [32, 19] C [3] 1 4 ≥ C × ( C ⋊ C ) [32, 22] { e } [1] 3 4 C ≀ C [32, 27] { e } [1], C [46] 3, 1 4, 3 C [7] 2 4 ≥ C ⋊ C [32, 33] C [5] 2 3 C ⋊ D [32, 34] C [5] 2 4 ≥ C × D [32, 39] C [3] 2 4 ≥ C × QD [32, 40] C [3] 2 4 ≥ C ⋊ C [32, 43] C [3], C [28] 2, 1 4, 3 ≥ ≥ C × D [32, 46] C [3] 3 4 4 D ◦ D [32, 49] C [32] 2 3 3 References [AL20] G.Arone and K.Lesh, Fixed points of coisotropic subgroups of Γ k on decompositionspaces , Homotopy Homotopy Appl. 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