On the RO(G) -graded coefficients of dihedral equivariant cohomology
OON THE RO ( G ) -GRADED COEFFICIENTS OFDIHEDRAL EQUIVARIANT COHOMOLOGY IGOR KRIZ AND YUNZE LU
Abstract.
We completely calculate the RO ( G ) -graded coefficientsof ordinary equivariant cohomology where G is the dihedral groupof order 2 p for a prime p > p = p . These are the first suchcalculations for a non-abelian group. Introduction
A 1982 Northwestern conference problem asked for a complete cal-culation of the RO ( G ) -graded cohomology groups of a point for a non-trivial finite group G (see [12] for definitions and [11] for background).This question was quickly solved by Stong [14] for cyclic groups Z / p with p prime. Partial calculations for groups Z /( p n ) and ( Z / p ) n weremuch more recently done in [4, 7, 5, 6, 9]. In a recent lecture, PeterMay [13] emphasized the fact that no case of a non-abelian group wasknown to date.The purpose of this paper is to advance progress in the non-abeliandirection by calculating the RO ( G ) -graded cohomology coefficients for G = D p , dihedral group with 2 p elements for p a prime number, withboth Burnside ring A and constant Z coefficients. The constant mackeyfunctor Z is obtained by taking the quotient of the Burnside ringMackey functor A by its augmentation ideal. Burnside Mackey functoris universal among among ordinary RO ( G ) -graded cohomology theo-ries in the same sense as Z -coefficients are non-equivariantly (see [3]),and thus were of primary interest historically. However, for non-trivialgroups, the Burnside ring is not a regular ring, and because of that,passage from Burnside ring to other coefficients is not immediate. Inapplications [8, 4], the use of constant coefficients, which are simpler,prevailed so far. Kriz acknowledges the support of a Simons Collaboration Grant. a r X i v : . [ m a t h . A T ] M a y IGOR KRIZ AND YUNZE LU
Our main tool is using an explicit D p -equivariant CW structure onrepresentation spheres, which will be described in the next section.We will state the calculation with constant Z coefficients here, andpostpone the statement with the Burnside ring coefficients till Section4 below, as it essentially follows from the constant case after somealgebraic examinations of the Burnside rings.We present G = D p as { ζ, τ ∣ ζ p = , τ = , ζτ = τ ζ − } . The group G has two one-dimensional representations: the trivial rep-resentation denoted by (cid:15) and the sign representation denoted by α .The group G also admits p − two-dimensional representations, denotedby γ i ’s, given by γ i ∶ ζ ↦ [ cos ( πip ) − sin ( πip ) sin ( πip ) cos ( πip ) ] , τ ↦ [ − ] , ≤ i ≤ p − . We will prove a periodicity result that will exempt us from distinguish-ing different two-dimensional representations. Hence the cohomologycould be indexed by k(cid:15) + (cid:96)α + mγ .To discuss the Z -coefficient case, it is useful to recall the followingcalculation due to Stong (see [10, 8]). Denote, for (cid:96) ≥ B (cid:96) = ̃ H D p ∗ ( S (cid:96)α , Z ) = ̃ H Z / ∗ ( S (cid:96)α , Z ) , (2) B (cid:96) = ̃ H ∗ D p ( S (cid:96)α , Z ) = ̃ H ∗ Z / ( S (cid:96)α , Z ) . Proposition 1.
Let n denote the grading. We have B (cid:96),n = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ Z n = (cid:96) even Z / ≤ n < (cid:96) even else, B (cid:96),n = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ Z n = (cid:96) even Z / ≤ n ≤ (cid:96) odd else. ◻ We also put B (cid:96),n = B − (cid:96), − n , B (cid:96),n = B − (cid:96), − n for (cid:96) < . Then (1) and (2) extend to (cid:96) < QUIVARIANT DIHEDRAL COHOMOLOGY 3
Now define s A t and s A t by(3) ( s A t ) n = { Z / p when 2 s < n < t − n ≡ , ( s A t ) n = { Z / p when 2 s < n < t − n ≡ , RO ( G ) -graded (co)homology of a point withcoefficients in Z is given by the following Theorem 2.
For m > , we have (5) H D p ∗ ( S mγ + (cid:96)α , Z ) = (cid:96) − A (cid:96) + m [− (cid:96) + ] ⊕ B (cid:96) + m [ m ] , (6) H ∗ D p ( S mγ + (cid:96)α , Z ) = (cid:96) A (cid:96) + m [− (cid:96) + ] ⊕ B (cid:96) + m [ m ] . Here [ k ] denotes shift up by k in homology or cohomology. Notethat since it is often appropriate to identify the cohomological gradingwith the nagative of homological, some authors prefer to define shiftsin one grading only; in that case, there would be a negative sign in thesquare brackets of one of the formulas (5), (6).Theorem 2 and Proposition 1 give a complete calculation of the RO ( G ) -graded cohomology of a point with Z coefficients. We willprove Theorem 2 in Section 2, 3 below, and give the discussion ofBurnside ring coefficients in Section 4.2. Equivariant CW-structure and periodicity
We will write G = D p from now on. By abuse of notation, in additionto the generator of D p , τ will also denote complex conjugation. Also,we shall write γ = γ .Let S ( mγ i ) be the unit sphere of the representation mγ i . In this sec-tion we will construct a D p -equivariant CW structure on each S ( mγ i ) .By computing the associated Mackey functor-valued equivariant chaincomplexes (meaning the Mackey functor-valued chain complexes givenby the fixed points of the cellular chain complex of the equivariantCW-complex with respect to subgroups) for different γ i ’s, we provethat instead of indexing on all γ i ’s, it suffices to consider only γ .The CW structure is obtained by subdividing the standard Z / p -equivariant cells of S ( mγ i ) . We will identify the nonequivariant under-lying spaces of all S ( mγ i ) ’s with subsets of C m (by identufying eachcopy of γ i with a copy of C ). Then for S ( mγ i ) , ζ ∈ G simply acts IGOR KRIZ AND YUNZE LU by coordinate-wise ζ ip multiplication where ζ p = e πi / p . In this con-text we will see S ( mγ i ) ’s share exactly the same CW decompositionnon-equivariantly, with different D p -actions.First observe that the usual free Z / p -equivariant CW-sructure on S ( mγ i ) has equivariant cells freely generated by the following non-equivariant cells for 1 ≤ k ≤ m :(7) {( z , ..., z k , , ..., ) ∈ S ( mγ i ) ∣ z k ∈ [ , ]} , (8) {( z , ..., z k , , ..., ) ∈ S ( mγ i ) ∣ z k ∈ [ , ]⋅ e λi , ( p − ) π / p ≤ λ ≤ ( p + ) π / p } . Though both (7) and (8) are stable under the action of τ , they are not D p -cells since τ acts non-trivially on them, and not all points of eachcorresponding open cell have the same isotropy. However it is worthnoting that they can be identified with unit disks of the representations(9) ( k − ) α + ( k − ) (cid:15), kα + ( k − ) (cid:15), respectively. This gives a guide on how to subdivide them into D p -equivariant cells. To be precise, we consider the following cells for S ( mγ i ) : Type A. a k,(cid:96) , ≤ (cid:96) ≤ k − , ≤ k ≤ m, generated by {( z , ..., z k , , ..., ) ∈ S ( mγ i ) ∣ Im ( z (cid:96) ) ≥ , z (cid:96) + , ..., z k − ∈ [− , ] , z k ∈ [ , ]} . The cell a k,(cid:96) has dimension k + (cid:96) − Z / (cid:96) = { e } for (cid:96) > Type B. b k,(cid:96) , ≤ (cid:96) ≤ k − , ≤ k ≤ m, generated by {( z , ..., z k , , ..., ) ∈ S ( mγ i ) ∣ Im ( z (cid:96) ) ≥ , z (cid:96) + , ..., z k − ∈ [− , ] , z k ∈ [− , ]} . Since it is symmetric to a k,(cid:96) , the cell b k,(cid:96) has dimension k + (cid:96) − Z / (cid:96) = { e } for (cid:96) > Type C. c k , ≤ k ≤ m, generated by {( z , ..., z k , , ..., ) ∈ S ( mγ i ) ∣ z k ∈ [ , ] ⋅ e iλ , ≤ λ ≤ π / p } . The cell c k has dimension 2 k − { e } . QUIVARIANT DIHEDRAL COHOMOLOGY 5
It is straightforward to check that these cells give a D p -equivarantCW decomposition for each S ( mγ i ) , only with different D p -actions fordifferent S ( mγ i ) ’s.Based on the equivariant CW-structure, we are ready to write downthe differentials. Note that the CW-structure is regular: the boundariesof cells attach by homeomorphic embeddings, and hence the nonzerocoefficients of the differentials will always be either 1 or −
1. We orientall cells as submanifolds (with corners) of the complex vector space C m . The induced orientation of the boundary of a cell is chosen bythe following rule: the induced orientation followed by the outwardnormal direction together make up the standard orientation of C m .For example, the induced orientation of S ⊂ C is going clockwise,hence the incidence number between a , and c is − Lemma 3.
Given ≤ i ≤ ( p − )/ , let ≤ j ≤ p − be the multiplica-tive inverse of i . Let ζ i = ζ j . With respect to the CW-structure andorientations described above, the D p -equivariant cell chain complex of S ( mγ i ) in the sense of Bredon [2] has differential da , = db , = dc = ζ p + i b , − a , da , = − a , − ( + ζ i + ... + ζ p − i ) c + ( ζ i + ... + ζ p − i ) τ c db , = − b , − ( + ζ i + ... + ζ p − i ) c + ( ζ i + ... + ζ p − i ) τ c da k, = a k − , − b k − , k > db k, = a k − , − b k − , k > da k, = a k − , − b k − , + (− ) k − a k, k > db k, = a k − , − b k − , + (− ) k − b k, k > k > , < (cid:96) < k − da k,(cid:96) = a k − ,(cid:96) − b k − ,(cid:96) + (− ) k − (cid:96) a k,(cid:96) − + (− ) k − τ a k,(cid:96) − db k,(cid:96) = a k − ,(cid:96) − b k − ,(cid:96) + (− ) k − (cid:96) b k,(cid:96) − + (− ) k − τ b k,(cid:96) − For k >
2, by abbreviating the action of ∑ ( p − )/ j = ζ ji to σ , da k,k − = − a k,k − + (− ) k − τ a k,k − − ( + σ ) c k − + (− ) k − στ c k − db k,k − = − b k,k − + (− ) k − τ b k,k − − ( + σ ) c k − + (− ) k − στ c k − Finally, for k > dc k = − a k,k − + (− ) k τ a k,k − + ζ p + i b k,k − + (− ) k − ζ p + i τ b k,k − . Proof.
We present here a computation for the differential of a k,k − for k >
2. By equivariance, it suffices to work on the generator, which is
IGOR KRIZ AND YUNZE LU given by {( z , ..., z k , , ..., ) ∈ S ( mγ i ) ∣ Im ( z k − ) ≥ , z k ∈ [ , ]} . Note that z k is uniquely determined by the values of z , ..., z k − , andthe dimension of the cell is 2 k −
2. Hence we only need to consider cellsof dimension 2 k − a k,k − attaches. They are precisely thosecells with z k − coordinates lying on the boundary of a k,k − , namely, a k,k − , τ a k,k − , c k − , ζ i c k − , ..., ζ ( p − )/ i c k − , ζ i τ c k − , ..., ζ ( p − )/ i τ c k − . Here cells in the orbit of c k − are those with Im ( z k − ) ≥ a k,k − andthese cells. By the rule set above, we could use the basis(10) ( e , ie , e , ie , ..., e k − , ie k − ) to determine the orientation of a k,k − , and the orientation of τ a k,k − could be described by(11) ( e , − ie , e , − ie , ..., e k − , − ie k − , e k − ) . On a point of τ a k,k − that a k,k − attaches, the induced orientation isgiven by(12) ( e , ie , ..., e k − , ie k − , − e k − ) since juxtaposing with outward normal direction − ie k − gives the sameorientation as (10). It is straightforward to compare orientations (11)and (12) and this gives the sign da k,k − = ... + (− ) k − τ a k,k − + ... in the formula. All the other computations follow by direct inspectionin a similar way. ◻ Since S mγ i is the unreduced suspension of S ( mγ i ) , the D p -equivariantCW structure of S mγ i is easily derived.We will next prove that the choice of two-dimensional representation γ i doesn’t matter in the computation of ordinary equivariant cohomol-ogy. Let A denote the Burnside ring Green functor (see [3]). Proposition 4.
Let M be a D p -Mackey functor. The D p -stable ho-motopy type of the HA -module spectrum HM ∧ S γ i does not depend onthe choice of i . QUIVARIANT DIHEDRAL COHOMOLOGY 7
The proof of this result will occupy the remainder of this section.Let M be a Mackey functor. Generally, if X is a finite G -CW complex,write X n / X n − = X n + ∧ S n where X n is the n th skeleton and X n is a discrete G -set. We have achain complex of Mackey functors C ∗ ( X ; M ) given by C n ( X ; M ) = π ( HM ∧ X n + ) . It is also true that for any finite G -set S , C n ( X ; M )( S ) = M ( S × X n ) , which is the associated Mackey functor, also denoted by M X n , to afinite G -set X n .To compute the D p -Mackey functor-valued chain complex C ∗ ( S γ i ; M ) for constant coefficient Z and Burnside coefficient A , we start with de-scribing some D p -Mackey functors. Despite the fact that the group D p is non-abelian, its conjugacy relations among subgroups are sim-ple and we can depict a D p -Mackey functor M by a diagram of thefollowing form: M ( D p / e ) (cid:32) (cid:32) (cid:119) (cid:119) M ( D p /⟨ τ ⟩) (cid:55) (cid:55) (cid:32) (cid:32) M ( D p /⟨ ζ ⟩) (cid:96) (cid:96) (cid:119) (cid:119) M ( D p / D p ) (cid:96) (cid:96) (cid:55) (cid:55) Example 5.
Constant Mackey functor Z . Z p (cid:22) (cid:22) (cid:121) (cid:121) Z (cid:56) (cid:56) p (cid:22) (cid:22) Z (cid:85) (cid:85) (cid:121) (cid:121) Z (cid:85) (cid:85) (cid:56) (cid:56) Example 6.
Given a Z [ G ] -module M , we have fixed-point Mackeyfunctor M defined by M ( G / H ) = M H , restriction given by inclusion,and transfer given by summing over cosets. For example the fixed point IGOR KRIZ AND YUNZE LU
Mackey functor Z [ D p /⟨ τ ⟩] is given by Z [ D p /⟨ τ ⟩] ( , ,..., ) (cid:33) (cid:33) B (cid:118) (cid:118) p − Z ⊕ Z A (cid:54) (cid:54) ( ,..., , ) (cid:34) (cid:34) Z [ , ,..., ] (cid:98) (cid:98) (cid:115) (cid:115) Z [ , ,..., ] (cid:99) (cid:99) (cid:51) (cid:51) Here round brackets stand for row vectors while square brackets standfor column vectors, and A = ⎡⎢⎢⎢⎢⎢⎣ I I p − J p − ⎤⎥⎥⎥⎥⎥⎦ , B = [ I p − J p − ] . where I n is the n × n identity matrix and J n is the n × n minor diagonalidentity matrix.Similarly, the fixed point Mackey functor Z [ D p / e ] is given by thefollowing diagram. Z [ D p / e ] D (cid:34) (cid:34) ( I p ,I p ) (cid:116) (cid:116) p Z [ I p ,I p ] (cid:54) (cid:54) ( , ,..., ) (cid:34) (cid:34) Z [ D p /⟨ ζ ⟩] C (cid:98) (cid:98) ( , ) (cid:115) (cid:115) Z [ , ,..., ] (cid:98) (cid:98) [ , ] (cid:54) (cid:54) where the matrices are represented by C = ([ , ..., , , ..., ] , [ , ..., , , ..., ]) ,D = [( , ..., , , ..., ) , ( , ..., , , ..., )] . The matrices above are derived by arranging the order of cells care-fully: The basis of Z [ D p /⟨ τ ⟩] can be identified with cells generatedby a , . Recalling that ζ i acts by 2 π / p -rotation, we put a geometriccounterclockwise order on the cells a , , ζ i a , , ..., ζ p − i a , . We also put an order on the generators of Z [ D p /⟨ τ ⟩] ⟨ τ ⟩ by ζ i a , + ζ p − i a , , ..., ζ p − i a , + ζ p + i a , , a , , and this is why the upper left pair of arrows in the diagram for Z [ D p /⟨ τ ⟩] has the given matrix representation. QUIVARIANT DIHEDRAL COHOMOLOGY 9
The basis of Z [ D p / e ] can be identified with cells generated by c .We arrange them in the following order: c , ζ i c , ..., ζ p − i c , τ c , τ ζ i c , ..., τ ζ p − i c . The fixed point submodules are endowed with the induced order ofbasis.Now fix M = Z . In this case, by the double coset formula, theassociated chain complex of Mackey functors can be calculated as fixedpoint Mackey functos. Hence using the examples above, the Mackeyfunctor-valued D p -equivariant chain complexes for S γ i is the following: Z ←— Z [ D p /⟨ τ ⟩] ⊕ Z [ D p /⟨ τ ⟩] ←— Z [ D p / e ] . The differentials are derived from Lemma 3. Since the differentials are D p -equivariant, we immediately see that all chain complexes for thedifferent S γ i ’s are isomorphic.However, the isomorphism is not induced by any D p -equivariantmap between the representation spheres. To prove Proposition 4 weinstead want to construct a functor H ∶ Ch ≥ ( M ack ) → D S p G suchthat(1). H M = HM .(2). H C ∗ ( X ; M ) ≃ X ∧ HM . Construction:
Let H be the composition of the following functors Ch ≥ ( M ack ) K —→ sM ack H —→ sD S p G ∣⋅∣ —→ D S p G where K is the functor in Dold-Puppe correspondence which is anequivalence of first two categories; H is the Eilenbeg-Maclane func-tor and ∣ ⋅ ∣ is geometric realization functor. The Eilenberg-Maclaneconstruction is functorial; a recent account of this is in [1].As an example we compute the case when X = G / H + . Then C ∗ ( X ; M ) is concentrated on degree 0. All the functors are computable, and wehave H C ∗ ( X ; M ) = HM G / H ≃ HM ∧ G / H + . The last equivalence can be verified by computing the homotopy groupsof HM ∧ G / H + , and using the uniqueness of Eilenberg-MacLane spectra.In fact, one could make it into an natural isomorphism. By theprojection formula G ⋉ H HM ≅ G / H + ∧ HM and adjunction, it arises from the natural map of H -spectrum HM → HM G / H induced by inclusion at the coset eH : M ↪ M G / H . For any finite G -CW complex X , we realize it as a simplicial G -setand the functor H is constructed as above. Then Proposition 4 followsdirectly. 3. Proof of Theorem 2
In this section we still focus on Z Mackey functor coefficient andwill present a proof of Theorem 2. To do this, first we calculatethe D p -equivariant homology and cohomology of S ( mγ ) as Bredon(co)homology. Recall that there is a cellular filtration on S ( mγ ) bythe Z / p -equivariant cells generated by (7), (8) of dimension ≤ s . For k ≥
1, the filtration degree 2 k − b k,(cid:96) , c k andthe degree 2 k − a k,(cid:96) . By using the differen-tials computed above, the corresponding homological spectral sequencehas the following E -term: E k − , ∗ = B k − [ k − ] , for 1 ≤ k ≤ mE k, ∗ = B k [ k − ] , for 1 ≤ k ≤ m. The nontrivial differential d is also determined by Lemma 3, which isan isomorphism except for E j, → E j − , ∶ Z p —→ Z . On the two verti-cal edges s = , m , the terms also survive and the spectral sequencecollapses to the E page. In the case of cohomology, one just needs toturn subscripts into superscipts, mirror the computations by reversingarrows and use restriction maps of Mackey functors. Thus, we haveproved the following Proposition 7.
For m > , we have H D p ∗ ( S ( mγ ) , Z ) = Z ⊕ A m ⊕ B m [ m − ] ,H ∗ D p ( S ( mγ ) , Z ) = Z ⊕ A m ⊕ B m [ m − ] . ◻ It may be tempting to try to use the same method for calculatingthe reduced D p -equivariant (co)homology of Σ (cid:96)α ∧ S ( mγ ) + for (cid:96) ∈ Z ,but there are two difficulties. First, for (cid:96) >
0, the chain complex weobtain by smashing the CW-complexes cell-wise grows with (cid:96) . Moreimportantly, for (cid:96) <
0, the method actually fails: the Bredon chain com-plex is not an equivariantly stable object, and actually does not exist
QUIVARIANT DIHEDRAL COHOMOLOGY 11 for spectra obtained by desuspending by non-trivial representations.There is, of course, a concept of an equivariant CW-spectrum [11], butany chain complex in this stable context has to be built directly on theMackey functor level.We proceed as follows: Suspend the filtration above by S (cid:96)α . Thecorresponding spectral sequence’s d is determined by S (cid:96)α -suspensionof the connecting map F k + / F k + → Σ F k + / F k of the triad ( F k + , F k + , F k ) . (Note: those are “odd-to-even” connecting maps; by Lemma 3, the“even-to-odd” connecting maps are 0.) Stably it does not depend on (cid:96) . Note that the filtration quotients have the following form: F k + / F k ≅ D p ⋉ Z / S k +( k + ) α , F k + / F k + ≅ D p ⋉ Z / S ( k + )+( k + ) α . Hence the connecting map is a stable D p -equivariant map D p /( Z / ) + → D p /( Z / ) + By adjunction, it is equivalent to a Z / S → D p /( Z / ) + which is classified by an element in(13) A ( Z / ) ⊕ Z ⊕( p − )/ by the Wirthm¨uller isomorphism (which allows us to switch the sourceand target) and the fact that Z / D p /( Z / ) is a disjoint union of one fixed point and ( p − )/ a k + ,k to c k ,namely from da k,k − = − a k,k − + (− ) k − τ a k,k − − ( + σ ) c k − + (− ) k − στ c k − . This shows that the connecting map does not depend on k , and is in(13) represented by the element ( , , . . . , ) . This corresponds to multiplication by p on the constant Mackey functor Z . It is convenient then to look at the spectral sequence of Σ (cid:96) + (cid:96)α S ( mγ ) + ,whose E page is a shift of the conjunction of both cohomology andhomology E page for S ( mγ ) , and it also collapses to the E -page.Thus, we obtain Proposition 8.
For m > , we have H D p ∗ ( Σ (cid:96)α S ( mγ ) + , Z ) = B (cid:96) ⊕ B (cid:96) + m [ m − ] ⊕ (cid:96) A (cid:96) + m [− (cid:96) ] ,H ∗ D p ( Σ (cid:96)α S ( mγ ) + , Z ) = B (cid:96) ⊕ B (cid:96) + m [ m − ] ⊕ (cid:96) A (cid:96) + m [− (cid:96) ] . ◻ Proof of Theorem 2:
We use the cofiber sequenceΣ (cid:96)α S ( mγ ) + → S (cid:96)α → S (cid:96)α + mγ to finish our computation. Looking at the long exact sequence in ho-mology, the morphism B (cid:96) → B (cid:96) is the transfer map p , which is an iso-morphism except in the top dimension when (cid:96) is even, and this givesan extra Z / p . Besides, all the other components are shifted up by 1.Hence we have proved that H D p ∗ ( S (cid:96)α + mγ , Z ) = B (cid:96) + m [ m ] ⊕ (cid:96) − A (cid:96) + m [− (cid:96) + ] . In cohomology the restriction maps always give isomorphisms, hence H ∗ D p ( S (cid:96)α + mγ , Z ) = B (cid:96) + m [ m ] ⊕ (cid:96) A (cid:96) + m [− (cid:96) + ] . ◻ Example 9.
As an example, we illustrate how to compute H D p ∗ ( Σ − α S ( γ ) + , Z ) . First we compute the D p -equivariant homology and cohomology of S ( γ ) . The following is the E page of the homological spectral se-quence for H D p ∗ ( S ( γ ) , Z ) . QUIVARIANT DIHEDRAL COHOMOLOGY 13 E page for H D p ∗ ( S ( γ ) , Z ) − − − − − Z Z Z
Z Z Z / Z / Z / Z / Z / Z / Z / Z / Z / Z / Z / Z / Z / Z / Z / pp pp The differential d is a multiplication by p when there is a Z in thetarget (which is supported by c k , k even). The exception is filtrationdegree 2 m − =
9, where there is no differential with that target, andfiltration degree 0, where there is no differential with that source. Thereis no room for higher differentials for dimensional reasons. Hence thespectral sequence collapses to the E page. The two vertical edges andthe t = E page of the cohomological spectral sequence. E page for H ∗ D p ( S ( γ ) , Z ) − − Z Z Z Z / Z / Z Z Z / Z / Z / Z / pp pp Now let us suspend by − − α . Since the filtration on S ( γ ) is givenby S , S α , S + α , ..., S + α , S + α , the filtered quotients are given by S − − α , S − − α , S − − α , ..., S − , S , S α . The following is the E page, which is a shift of a juxtaposition of thedual of a truncation (at filtration degree 7) of the cohomological E page and a truncation (at filtration degree 1) of the homological E page. E page for ˜ H D p ∗ ( Σ − − α S ( γ ) + , Z ) −
101 0
ZZZ / ZZ Z / Z / ZZ / ppppE page for ˜ H D p ∗ ( Σ − − α S ( γ ) + , Z ) −
101 00 Z / p Z / p Z / ZZ / Burnside ring coefficients
In this section we will apply the procedure above to deal with theBurnside ring coefficient A . The difference is somewhat minor here dueto the fact that most of the cells are free and therefore the computationdiffers only in a few dimensional degrees. We denote the Burnsiderings additively by A ( Z / ) = Z { , t } , A ( Z / p ) = Z { , t p } , A ( D p ) = QUIVARIANT DIHEDRAL COHOMOLOGY 15 Z { , t , t p , t p } where t i denotes the orbit of cardinality i . The BurnsideMackey functor A is depicted as Z ↦ t p (cid:34) (cid:34) ↦ t (cid:118) (cid:118) Z { , t } t ↦ (cid:52) (cid:52) ↦ t p ,t ↦ t p (cid:33) (cid:33) Z { , t p } t p ↦ p (cid:97) (cid:97) ( ,t p )↦( t ,t p ) (cid:118) (cid:118) Z { , t , t p , t p } t p ↦ + p − t ,t p ↦ pt (cid:97) (cid:97) t p ↦ t p (cid:54) (cid:54) Now we compare the associated Mackey functor-valued chain com-plex for different S γ i ’s. By remembering the isotropy in the Z case, wesimply replace Z by the corresponding Burnside ring. For example theMackey functor A D p /⟨ τ ⟩ is p Z ↦ t p (cid:30) (cid:30) ↦ t (cid:119) (cid:119) A ( Z / ) ⊕ p − Z t ↦ (cid:53) (cid:53) ↦ t p ,t ↦ t p (cid:33) (cid:33) Z ≅ A ( ) t p ↦ p (cid:94) (cid:94) ( ,t p )↦( t ,t p ) (cid:120) (cid:120) A ( Z / ) t p ↦ + p − t ,t p ↦ pt (cid:97) (cid:97) t p ↦ t p (cid:56) (cid:56) Since all restriction and transfer maps in the associated Mackey func-tors are induced from the ones in A , if we again order equivariant cellscarefully, then the maps are the same. The differential is also inducedfrom equivariant differential on the cellular level, so the chain com-plexes A ←— A D p /⟨ τ ⟩ ⊕ A D p /⟨ τ ⟩ ←— A D p /⟨ e ⟩ are all equivalent, and we also have periodicity for the Burnside ringcoefficients.Denote, for (cid:96) ≥ B (cid:96) = ̃ H Z / ∗ ( S (cid:96)α , A ) , (15) B (cid:96) = ̃ H ∗ Z / ( S (cid:96)α , A ) . Denote by I Z / the kernel of the restriction A ( Z / ) → A ( ) , and by J Z / the cokernel of the induction A ( ) → A ( Z / ) . Both I Z / and J Z / are clearly isomorphic to Z . We can repeat the computations via spectralsequences to obtain the following Proposition 10. (Stong [10, 8] ) For (cid:96) ≥ , we have B (cid:96),n = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ J Z / n = Z n = (cid:96) even Z / < n < (cid:96) even else, B (cid:96),n = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ I Z / n = Z n = (cid:96) even Z / ≤ n ≤ (cid:96) odd else. ◻ By setting B (cid:96),n = B − (cid:96), − n , B (cid:96),n = B − (cid:96), − n for (cid:96) < (cid:96) < Proposition 11.
For m > , we have H D p ∗ ( S ( mγ ) , A ) = A ( Z / ) ⊕ A m ⊕ B m [ m − ] ,H ∗ D p ( S ( mγ ) , A ) = A ( Z / ) ⊕ A m ⊕ B m [ m − ] . ◻ To compute suspensions by (cid:96)α , we need to see what the attachingfmap D p /( Z / ) + → D p /( Z / ) + induces on A . In terms of Z / S → D p /( Z / ) + , it induces A ( Z / ) → A ( Z / ) ↦ + p − t , t ↦ pt . Thus, it is injective, and its cokernel is Z / p , just as with Z coefficients.Hence QUIVARIANT DIHEDRAL COHOMOLOGY 17
Proposition 12.
For m > , we have H D p ∗ ( Σ (cid:96)α S ( mγ ) + , A ) = B (cid:96) ⊕ B (cid:96) + m [ m − ] ⊕ (cid:96) A (cid:96) + m [− (cid:96) ] ,H ∗ D p ( Σ (cid:96)α S ( mγ ) + , A ) = B (cid:96) ⊕ B (cid:96) + m [ m − ] ⊕ (cid:96) A (cid:96) + m [− (cid:96) ] . ◻ Finally, to get the (co)homology of S (cid:96)α + mγ , we look at the cofibersequence again. Denote(16) C (cid:96) = ̃ H D p ∗ ( S (cid:96)α , A ) , (17) C (cid:96) = ̃ H ∗ D p ( S (cid:96)α , A ) . Also denote by J Z / pD p the cokernel of the induction A ( Z / p ) → A ( D p ) ,and by I Z / pD p the kernel of the restriction A ( D p ) → A ( Z / p ) . Both areisomorphic to Z ⊕ Z as groups. Since the Weyl group of Z / ⊂ D p is itself and it acts trivially on the Burnside ring A ( Z / p ) , spectralsequence computation shows that Proposition 13.
For (cid:96) ≥ , we have C (cid:96),n = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ J Z / pD p n = A ( Z / p ) n = (cid:96) even A ( Z / p )/ < n < (cid:96) even else, C (cid:96),n = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ I Z / pD p n = A ( Z / p ) n = (cid:96) even A ( Z / p )/ ≤ n ≤ (cid:96) odd else.For (cid:96) < , put C (cid:96),n = C − (cid:96), − n , C (cid:96),n = C − (cid:96), − n . ◻ Note that we have short exact sequences0 → B (cid:96) ind Z / D p ———→ C (cid:96) → B (cid:96) → , and 0 → B (cid:96) → C (cid:96) res D p Z / ———→ B (cid:96) → . Therefore
Theorem 14.
For m > , we have H D p ∗ ( S mγ + (cid:96)α , A ) = B (cid:96) ⊕ (cid:96) − A (cid:96) + m [− (cid:96) + ] ⊕ B (cid:96) + m [ m ] ,H ∗ D p ( S mγ + (cid:96)α , A ) = B (cid:96) ⊕ (cid:96) A (cid:96) + m [− (cid:96) + ] ⊕ B (cid:96) + m [ m ] . ◻ References [1] A.Bohmann, A.Osorno: Constructing equivariant spectra via categorialMackey functors,
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