COMPUTING THE MORAVA K-THEORY OF REAL GRASSMANIANS USING CHROMATIC FIXED POINT THEORY.

. We study K ( n ) ∗ (Gr d ( R m )), the 2-local Morava K -theories of the real Grassmanians, about which very little has been previously computed. We conjecture that the Atiyah-Hirzebruch spectral sequences computing these all collapse after the ﬁrst possible non-zero diﬀerential d 2 n +1 − 1 , and give much evidence that this is the case. We use a novel method to show that higher diﬀerentials can’t occur: we get a lower bound on the size of K ( n ) ∗ (Gr d ( R m )) by constructing a C 4 –action on our Grassmanians and then applying the chromatic ﬁxed point theory of the authors’ previous paper. In essence, we bound the size of K ( n ) ∗ (Gr d ( R m )) by computing K ( n − 1) ∗ (Gr d ( R m ) C 4 ). Meanwhile the size of E 2 n +1 is given by Q n –homology, where Q n is Milnor’s n th primitive mod 2 cohomology operation. Whenever we are able to calculate this Q n –homology, we have found that the size of E 2 n +1 agrees with our lower bound for the size of K ( n ) ∗ (Gr d ( R m )). We have two general families where we prove this: m ≤ 2 n +1 and all d , and d = 2 and all m and n . Computer calculations have allowed us to check many other examples with larger values of d .


Introduction
Let Gr d (R m ) be the real Grassmanian of k-planes in R m , a much studied compact manifold of dimension d(m − d) admitting the structure of a CW complex with m d 'Schubert cells'.Much is known about the ordinary cohomology of these spaces: (1) H * (Gr d (R m ); Z/2) is generated by Stiefel-Whitney classes satisfying standard relations.It has total dimension m d .
(2) H * (Gr d (R m ); Q) is generated by Pontryagin classes, along with, in some cases, an odd dimensional class.For fixed d, and ǫ = 0 or 1, the total dimension of H * (Gr d (R 2−ǫ+2l ); Q) is polynomial of degree ⌊d/2⌋ as a function of l ≥ 0. (3) If m is even, then Gr d (R m ) is oriented.Furthermore, the inclusion Gr d (R m−1 ) ֒→ Gr d (R m ) induces an epimorphism in rational cohomology.(4) Nontrivial torsion in H * (Gr d (R m ); Z) has order 2. The mod 2 Bockstein Spectral Sequence (BSS) collapses after the first differential.Equivalently, the mod 2 Adams Spectral Sequence (ASS) converging to H * (Gr d (R m ); Z) collapses at E 2 .Much less is known about other cohomology theories applied to these Grassmanians.In this paper, we study K(n) * (Gr d (R m )) for n ≥ 1.Here K(n) * (X) denotes the 2-local nth Morava K-theory of a space X, a graded vector space over the graded field K(n) * = Z/2[v ± n ] with |v n | = 2 n+1 − 2. We let k(n) denote the connective cover of K(n Viewing HQ as K(0) and HZ as k(0), our discovery is that analogues of statements (2)-(4) above appear to hold for all n, with the Atiyah-Hirzebruch Spectral Sequence (AHSS) replacing the Bockstein Spectral Sequence in statement (4).Furthermore, the analogue of statement (1) holds through a much bigger range than one would expect from dimension considerations.
In the next two subsections, we describe our main results.
Given a finite complex X and n ≥ 0, we let k n (X) = dim K(n) * K(n) * (X).
We note that this collapsing range is surprisingly large, as dimension considerations just imply collapsing if d(m − d) < 2 n+1 .
For larger m, we have the following lower bound.
The biggest novelty of this paper is our method for proving Theorems 1.1 and 1.2: we make use of chromatic fixed point theory to prove these nonequivariant results.
The blue shift theorem of [6A19] says that if C is a finite cyclic p-group and X is a finite C-CW complex, then In [KL20], we upgraded this as follows.
Theorem 1.4.[KL20,Theorem 2.16] If C is a finite cyclic p-group, and X is a finite C-CW complex, then (In these statements, K(n) * means Morava K-theory at the prime p.) As m d is an evident upper bound for k n (Gr d (R m )), to prove Theorem 1.1, it suffices to show that k n (Gr d (R m )) ≥ m d in the stated range.Using Theorem 1.4, we will show this by induction on n using a C 2 -action on Gr d (R m ) induced by an m-dimensional real representation of C = C 2 .
We will similarly prove Theorem 1.2 for n ≥ 1 by using a C 4 -action on Gr d (R m ) induced by an m-dimensional real representation of C = C 4 .
In both cases, it will be quite easy to compute k n−1 (Gr d (R m ) C ).Details of this will be in §2.
1.2.Results about the Q n -homology of the Grassmanians.
Conjecture 1.3 follows from a conjectural calculation that only involves H * (Gr d (R m ); Z/2), viewed as a module over the Steenrod algebra.
As will be reviewed in §3.1, the first differential in the AHSS converging to ).This makes it not hard to check the next lemma.
Lemma 1.5.If X is a finite complex, k Qn (X) ≥ k n (X) is always true, and the following are equivalent: We apply this to our situation.Firstly, Theorem 1.1 has the following nontrivial algebraic consequence.
(For an algebraic proof of this result using the methods of §3.5, see the second author's thesis [L21, p.75].) For m > 2 n+1 , we believe the following is true.
Known rational calculations imply that the conjecture is true when n = 0.It is also easy to show that the conjecture is true when d = 1, and one calculates With much more work we prove the following.

and we have the calculation
We are firm believers in our conjectures.For more evidence, the second author has made extensive computer calculations verifying Conjecture 1.7 in hundreds more cases with larger values of d: see the tables in the Appendix.
For d ≥ 2, computing the size of H * (Gr d (R m ); Q n ) seems tricky.We have organized our efforts by studying how these numbers change as m is increased as follows.
Let C d (R m ) denote the cofiber of the inclusion Gr ) is identified as the Thom space of the canonical normal bundle over Gr d−1 (R m−1 ), and in §3.4,we study the One has an induced short exact sequence of modules over the Steenrod algebra When m is even, we see much orderly behavior.
Theorem 1.9.Let m be even. (a) (c) If Conjecture 1.7 is true for (n, d, m − 1) and (n, d − 1, m − 1), then it is true for (n, d, m).Furthermore, Gr d (R m ) will then be k(n)-oriented, and the cofiber sequence above will induce short exact sequences We prove Theorem 1.9 in §4.We make use of the additive basis {s λ } dual to the classical Schubert cells.Here λ runs through partitions having at most d parts, each no bigger than m − d.In [L98], Cristian Lennart gave a combinatorial formula for Q n (s λ ), and we use this to prove (a).Duality statement (b) follows quite formally from (a), and (c) follows easily from (b).
When m is odd, the analogues of statements (a) and (b) are false, and, for d ≥ 3, the full behavior of the connecting map in the Q n -homology long exact sequence, is as yet unclear to the authors.In §6, we will prove analogues of Theorem 1.1 and Theorem 1.2 for C d (R m ), and then speculate on behavior of δ that would be compatible with all of our computations.
However, when d = 2, we have the following result.
Theorem 1.10.Let m > 2 n+1 be odd.Then k Qn (C 2 (R m )) = 2 n+1 − 2 and the map is zero, so that there is a short exact sequence From this Theorem 1.8 quickly follows and one can deduce that, in this case, there is a short exact sequence We prove Theorem 1.10 in §5.The tools we use are very different from those used in proving Theorem 1.9: we work with the classical presentation of H * (Gr d (R m ); Z/2)) as a ring of Stiefel-Whitney classes.

Comparison with other work.
When comparing our work to what has come before, the first thing to say is that the outcome of our calculations -though not the methods -are in line with the classical calculations first made by C.Ehresmann in 1937 [E37].He determined the additive structure of both H * (Gr d (R m ); Z/2) and H * (Gr d (R m ); Q).He also showed that all the torsion in H * (Gr d (R m ); Z) was of order 2; in modern terms this is equivalent to showing that the Bockstein spectral sequence computing H * (Gr d (R m ); Z) collapses after the first nonzero differential given by Q 0 = Sq 1 = β.
Calculating the Morava K-theories of Gr d (R ∞ ) = BO(d) was done first by Kono and Yagita [KY93], and then, with a simpler proof, by Kitchloo and Wilson [KW15].Again, the AHSS computing K(n) * (BO(d)) collapses after the first nonzero differential, but the collapsing is for an elementary reason: H * (BO(d); Q n ) is concentrated in even degrees.Indeed, one quickly learns that the complexification map BO(d) → BU (d) induces an epimorphism An equivalent statement is that H * (BO(d); Q n ) is generated by the classes w 2 1 , . . ., w 2 d .These will still be permanent classes in the AHSS converging to K(n) * (Gr d (R m )), but now we have odd dimensional classes as well, with the number of these seemingly growing as d and m grow.
Finally, we point out that we do not attempt to describe K(n) * (Gr d (R m )) as a K(n) * -algebra.Our results do tell us something about this, however.In the situation of Theorem 1.1, the known algebra H * (Gr d (R m ); Z/2)⊗ K(n) * will be an associated graded.Similarly, whenever our conjecture is valid, . What is still needed, and might be necessary to prove our conjectural collapsing in general, are sensible constructions of classes in odd degrees.
2. The proofs of Theorems 1.1 and 1.2 In this section we prove Theorems 1.1 and 1.2 by using our chromatic fixed point theorem Theorem 1.4.

A fixed point formula.
Let G be a finite group, and let V be an m-dimensional real representation of G. Then Gr d (V ), the space of d-planes in V , is a model for Gr d (R m ) with an evident G-action.Here we describe Gr d (V ) G , its space of G-fixed points.
To state this, we need some notation.Let V 1 , . . ., V k be the irreducible real representations of G, let r i = dim R V i , and let Each of the endomorphism algebras D i will be a finite dimensional real division algebra, and thus isomorphic to R, C, or H, and dim R D i will divide r i .
Proof.The fixed point space Gr d (V ) G will be the space of sub-G-modules W < V of real dimension d.Such a G-module W will decompose canonically as A submodule W i of V m i i must be isomorphic to V j i for some j, thus Gr d i (V m 1 i ) G will be empty unless d i = j i r i for some j i .Finally, using that Hom R 2.2.Proof of Theorem 1.1.
Using Theorem 1.4 and Proposition 2.1, we prove this by induction on n.
The n = 0 case of the theorem is easy to check as For the inductive step, assume that if Let C 2 be the cyclic group of order 2. To get our needed lower bound, our strategy will be to make R m into a C 2 -module, and then apply Theorem 1.4.
The group C 2 has two irreducible 1-dimensional real representations: call them L 1 and L 2 .Since m ≤ 2 n+1 , we can write m as m = p + q with both p ≤ 2 n and q ≤ 2 n .Now let Applying Proposition 2.1, we see that Applying Theorem 1.4 to this, we learn that Remark 2.4.The same inductive proof can be used to prove the classical result that dim Z/2 H * (Gr d (R m ); Z/2) = m d for all m and d, with our chromatic fixed point theorem Theorem 1.4 replaced by the classical theorem of Ed Floyd [F52,Theorem 4.4]: if the cyclic group C p acts on a finite CW complex X, then dim Z/p H * (X; Z/p) ≥ dim Z/p H * (X Cp ; Z/p).It would be interesting to know if this argument was known to Floyd, or others, like Bob Stong, who regularly worked with these sorts of group actions.

Proof of Theorem 1.2.
The strategy of the proof of Theorem 1.2 is the same as the proof in the last subsection: we get a lower bound on k n (Gr d (R m )) by letting a cyclic 2-group act on R m and applying Theorem 1.4.
In this case, the representation theory of C 2 is not rich enough to give us a big enough lower bound, but a well chosen real representation of the group C 4 of order 4 works better.Curiously, in our calculation of k n−1 of the resulting fixed point space, we are able to use our already proven Theorem 1.1, so the proof is not by induction, but more direct.
The group C 4 has three irreducible real representations: L 1 and L 2 of dimension 1, and Now let m = 2 n+1 − ǫ + 2l with ǫ = 0 or 1, and l ≥ 0. We define an m dimensional real representation Applying Theorem 1.4 to this, we learn that 3. The Q n homology of Gr d (R m ): background material 3.1.The AHSS and the ASS for Morava K-theory.
Let n ≥ 1.We recall the structure of the AHSS converging to K(n) * (X) (as always, in this paper, with p = 2).It is a spectral sequence of graded Sparseness of the rows implies that the differential d r will be zero unless r = s(2 n+1 −2)+1 for some s.The first possible nonzero differential, d 2 n+1 −1 , satisfies the following formula [Y80]: and so the dimension of E 2 n+1 (X) as a K(n) * -vector space will equal k Qn (X), the dimension of the Q n -homology of X.One immediately deduces part of Lemma 1.5: To continue with the proof of Lemma 1.5, let cE * ,⋆ r (X) denote the terms of the AHSS computing k(n) * (X), a 4th quadrant spectral sequence.Note and equals it for ⋆ ≤ 0, and that the latter spectral sequence is obtained from the former by inverting v n .It follows that cE * ,⋆ 2 n+1 (X) = E * ,⋆ 2 n+1 (X) for ⋆ < 0, with the map on the 0-line between the spectral sequences corresponding to the epimorphism From this, one sees that any higher differential in the k(n) * (X) AHSS would be detected in the K(n) * (X) AHSS.Since this 2nd spectral sequence is the localization of the first, we an conclude that the Next we note that the AHSS spectral sequence cE * ,⋆ r (X) identifies with the ASS computing k(n) * (X) with suitable re-indexing, with cE * ,⋆ 2 n+1 (X) corresponding to the Adams E 2 term.Firstly, a result of C.R.F.Maunder [M63] implies that the AHSS converging to [X, k(n)] * can be constructed by taking the Postnikov filtration of the spectrum k(n).But the Postnikov tower for k(n) is also an Adams tower: as described in in the survey paper [W91, §5], there is a cofibration sequence Finally, we note that, when n = 0, one still has the cofibration sequence as above, with now with v 0 = 2, so that the ASS for k(0) = HZ is similarly related to the Bockstein spectral sequence.

The description of H
We recall classical results that are either explicitly in [MS74] or can easily be deduced from the material there.
Let w 1 , . . ., w d denote the Stiefel-Whitney classes of the canonical ddimensional bundle γ d over Gr d (R ∞ ).One has Dual classes w1 , w2 , . . .are defined by the equation and this allows one to write the classes wk as polynomials in w 1 , . . ., w d .
The inclusion Gr d (R m ) ֒→ Gr d (R ∞ ) then induces a surjective ring homomorphism We record some useful consequences.To state these, it is useful to let be the inclusion induced by the inclusion R m−1 ֒→ R m , and to let Proof.Statement (a) follows from the recursive relations among the wk 's.Statement (b) follows from the equation , and (d) follows from (c), noting that j can be written as the composite where the indicated homeomorphisms are given by taking complimentary subspaces (and, in cohomology, these maps swap w i 's with wj 's).
We end this subsection with a couple more facts about H * (Gr d (R m ); Z/2).An additive basis for H q (Gr d (R m ); Z/2) is given by the monomials See [J89].The Wu formulae [MS74,p.94]are closed formulae for Sq i w j , and, in theory, formulae for Q n (w j ) follow.

3.3.
A description of the cofiber C d (R m ) and its cohomology.
Recall that C d (R m ) is defined as the cofiber of the inclusion Gr ).This cofiber can be identified as a Thom space as follows.
Proposition 3.2.Let S(γ ⊥ d−1 ) and D(γ ⊥ d−1 ) be the sphere and disk bundles associated to γ where e m is the mth standard basis vector in R m .We claim this f has the needed properties.Firstly, note that f (V, Finally, we need to check that f is bijective on and let v be the unique unit vector v ∈ W ∩ V ⊥ such that v has positive mth coordinate.Let π : R m → R m−1 be the standard projection.We claim that f (V, π(v)) = W and (V, π(v)) is the unique point in ) is a submodule.The proposition thus implies the following.
Proposition 3.4.Q 2 n = 0, and the chain complex )) be the isomorphism established in the last subsection: Θ(x) = x wm−d .The proposition follows once we check that Θ( Q n (x)) = Q n (Θ(x)).We compute: It is useful to put the class α n in context.Given any element a in the Steenrod algebra A, one gets a characteristic class w a (ξ) ∈ H |a| (B; Z/2) associated to any real vector bundle ξ → B: w a (ξ) is defined as the element satisfying a(u ξ ) = w a (ξ)u ξ ∈ H dim ξ+|a| (T h(ξ); Z/2), where u ξ is the Thom class of ξ.So, for example, w Sq n (ξ) = w n (ξ), and, relevant for us, our class ).We have the following characterization of w Qn .Proposition 3.5.w Qn is the unique characteristic class satisfying the following two properties: (a) w Qn (ξ ⊕ ν)) = w Qn (ξ) + w Qn (ν).
Proof.Property (a) follows from the fact that Q n is primitive in A (or, equivalently, that Q n acts a derivation).To see Property (b), one first calculates that Then Property (b) follows, since if γ is the universal line bundle over RP ∞ , then u γ = t.Uniqueness follows from the splitting principle.
Remark 3.6.Thus w Qn (ξ) agrees with the 's-class' s 2 n+1 −1 (ξ), analogous to the class of the same name for complex vector bundles as defined in [MS74,§16].(These s I 's are not the same as the s λ of the next subsection: these are two conflicting and standard usages.) 3.5.The description of H * (Gr d (R m ); Z/2) via Schubert cells, and Lenart's formula.
For the purposes of proving Theorem 1.9, we use an alternative description of H * (Gr d (R m ); Z/2).
We recall the cell structure of Gr d (R d+c ) as described in [MS74,§6].
The weight of λ is defined to be λ i and is denoted |λ|.Such a λ is a partition contained inside of a d × (m − d) grid when depicted as Young diagrams: diagrams with λ i boxes in the ith row.
To each such partition is associated a Schubert cell e(λ) of dimension λ in Gr d (R m ) defined by This cell decomposition of the Grassmanian leads to the dual Schubert cell basis for H * (Gr d (R m ); Z/2) with basis elements With this notation, one has that w i = s (1 i ) and wj = s ( j).Though we don't use this here, it is worth noting that the cohomology ring structure in this basis is described by the Littlewood-Richardson rule of symmetric function theory.
To state Lenart's formula for calculating Q n on a Schubert basis element [L98], we need some combinatorial definitions.Given a Young diagram λ that includes into another Young diagram µ, one can form the complement µ/λ.For example, The content of a box b of µ in row i and column j is defined to be c(b) = j − i.For a box b in the skew shape µ/λ, we define its content to be the content of b embedded in µ.Here we fill in the contents of the diagrams from above λ = 0 1 2 A skew-shape is said to be connected when each pair of boxes in the diagram is connected by a sequence of boxes that each share an edge.A shape λ is called a border strip, if it is connected and does not contain a 2 × 2 block of boxes.A shape satisfying just the second condition is called a broken border strip, and in particular, a border strip is an example of a broken border strip with just one connected component.If λ is a broken border strip, then we denote by cc(λ) the number of connected components of λ.If λ is not a broken border strip, then we define cc(λ) = ∞.For example, in the next diagram, λ 1 is a border strip, λ 2 is a broken border strip that is not a border strip, and λ 3 is an example of a shape that is neither.
A sharp corner of a broken border strip is a box with no north, no west and no northwest neighbors.A dull corner is a box with both north and west neighbors, but no northwest neighbor.Let C(µ/λ) denote the set of sharp and dull corners of µ/λ.For example, in the following diagram the sharp corners have been labeled S and the dull corners have been labeled D.

S S D
We are now ready to state Lenart's formula from [L98]: where µ/λ must be a broken border strip and Example 3.8.As an example we compute Q 1 on w 1 = s in the Schubert basis in Gr 2 (R 6 ).There are three basis elements in degree four To compute Q n (s ) using (3.1) we must consider each complement.Let λ = .For µ 1 we have The complement is a border strip and there is just one sharp corner (the left most corner) and no dull corners.The content of the sharp corner is 1 modulo two, hence d λµ 1 = 1, and so s µ 1 is in the expansion of Q 1 (s λ ).Next we consider This is a disconnected broken border strip, hence d λµ 2 = 1, and so s µ 2 is in the expansion.Finally, There are two sharp corners, one of content −1 and the other of content 1.
There is also one dull corner of content −2.This means d λµ 3 = (−1)+1+2 ≡ 0, and so s µ 3 is not in the expansion.Hence, Proof of Theorem 1.9(a).We are going to show that is given by (3.2).We must only consider λ such that (d (m−d) )/λ is a broken border strip.As (d (m−d) ) is a d × (m − d) grid the complement (d (m−d) )/λ is always connected and so if (d (m−d) )/λ is a broken border strip it must be, in particular, a border strip.If (d (m−d) )/λ is a border strip, then it must be one of three types: (1) (d (m−d) )/λ is the last row of (d )/λ is the union of the last row and last column of (d (m−d) ).We will show that d λ(d (m−d) ) = 0 in each of these cases.As m was assumed to be even, the content of the right most bottom box of (d (m−d) ) is also even.
(1) For the first case, there is just one sharp corner, namely the left most box, and there are no dull corners.Since the strip is of odd length, namely, 2 n+1 − 1, the left most box and the right most box have the same content modulo two.Hence, the content of this sharp corner is zero modulo two, and so d λ(d (m−d) ) = 0. (2) For the second case, the argument is exactly the same, but with the sharp corner on the top.(3) For the third case, the content of the sharp corner on the bottom left and the content of the sharp corner on the top right agree modulo two, because the border strip is of odd length.There is one dull corner in the bottom right and it is zero modulo two.Thus, the two sharp corners cancel and the dull corner contributes nothing.
Thus, in all cases Q n (s λ ) = 0 for s λ in degree m(m − d) − 2 n+1 + 1.This completes the proof that the top class is not in the image of Q n for even m.
Proof of Theorem 1.9(b).We wish to prove that, when m is even, then the chain complexes ( So we need to check that the chain complexes H * (Gr By Theorem 1.9(a), we know that and we conclude that 0 Proof of Theorem 1.9(c).Recall that k Qn (X) denotes the rank of the Q nhomology H * (X; Q n ).Similarly, let kQn (X) denote the rank of H * (X; Q n ).
Let m = 2 n+1 − ǫ + 2l with ǫ = 0 or 1, and l ≥ 0. Let We start with the first part of Theorem 1.9(c).This asserts that, when m is even, if we assume that with equality if and only if the associated long exact Q n -homology sequence is still short exact.
Since m is even, Theorem 1.9(b) applies, and tells us that kQn Putting this all together, under our assumptions, we have that d, m).That these would be, in fact, equalities, follows from the next lemma.
When all of this happens, we then see that the Q n -homology long exact sequence really is still short exact, and also that the K(n)-AHSS must collapse for these three spaces.Thus there is also a short exact sequence Finally, the top cohomology class in H d(m−d) (Gr d (R m ); Z/2) will be a permanent cycle in the AHSS computing K(n) * (Gr d (R m )) and thus also in the AHSS computing k(n) * (Gr d (R m )), and this is equivalent to saying that Gr d (R m ) is k(n)-oriented.

Results about H
In this section we present our results about the Q n -homology of Gr 2 (R m ), with the focus on understanding the case when m has the form 2 n+1 − 1+ 2l.
To begin with, we know the following: Lemma 5.1.In H * (Gr 2 (R ∞ ); Z/2) we have the following.
(a) w0 = 1, w1 = w 1 , and, recursively, wk = w Proof.The homogeneous components of the equation 0 = (1 + w 1 + w 2 )(1 + w1 + w2 + . . . ) give statement (a).Statement (b) is proved by induction on j.The case when j = 0 is trivial, and statement (a) rewrites as w 2 wk = w 1 wk+1 + wk+2 , which is the case when j = 1.One then computes For (c), note that wk is the homogeneous component of degree ≡ 1 mod 2 only if j = 0. Similarly, statement (e) follows from (c): Proof.The first equation here was already noted in the proof of Proposition 3.5, and the second follows from Lemma 5.1(d).
Proof.The second equation here follows from Lemma 5.1(e), so we just need to check the first.We do this by induction on n, where the n = 0 case is the easily checked: Before proceeding with the inductive step, we make two observations.The first is that for ) because the other term, Q n−1 Sq 2 n (w 2 ), will be zero.This is clear if n ≥ 2 as then Sq 2 n (w 2 ) = 0, and when n = 1, we observe that 2 ), and thus Now we check the inductive step of our proof.
As Q n is a derivation, the lemma, together with the calculation Q n (w 1 ) = w 2 n+1 1 , allows one to easily compute the Q n -homology of C 2 (R m ).What results is the following.
Proposition 5.6.(a) In H * (C 2 (R m ); Z/2), and is odd, then the classes Proof of Theorem 1.10.Let m = 2 n+1 + 1 + 2l.We need to prove that the map In other words, we need to show that representatives of the Q nhomology classes in H * (C 2 (R m ); Z/2) are in the image of Q n when regarded in H * (Gr 2 (R m ); Z/2).
By Proposition 5.6(d), these representatives are in two families: 1+2l , we will be done, as then Thus the next two propositions finish the proof.
We prove this by induction on m, with the two cases when l = 0 already covered by Theorem 1.1.The case when m is even is covered by Theorem 1.9(c), as we know our calculations are right for (n, 1, m − 1), and by induction we can assume the theorem for (n, 2, m − 1).
Proof of Proposition 5.7.We prove by induction on l that We start the induction by checking both the l = 0 and l = 1 cases.When l = 0, this reads Q n (w 1 ) = w 1 w2 n+1 , proved in Lemma 5.2.We check the l = 1 case using both Lemma 5.2 and Lemma 5.3: For the inductive case, we use the identity wk = w 2 2 wk−4 + w 2 1 wk−2 which holds for all k ≥ 4. Then we have Proof of Proposition 5.8.We wish to prove that We begin with a calculation in H * (Gr 2 (R ∞ ); Z/2): When we project this sum onto only the term with i = 0 is not zero.In other words holds in H * (Gr 2 (R 2 n+1 +1+2l ); Z/2).

Towards the conjectures
As organized in this paper, we are trying to calculate H * (Gr d (R m ); Q n ) by induction on m (and d) with two steps: ) is the Thom space of a bundle over Gr d−1 (R m−1 ), and When m is even, Theorem 1.9 says we can carry through with this plan.In this section we speculate about how things might go when m is odd.
Firstly, we have the analogues of Theorem 1.1 and Theorem 1.2 for kn Theorem 6.2.Let m = 2 n+1 − ǫ + 2l with ǫ = 0 or 1, and l ≥ 0. Then Proof.The proof is similar to the proof of Theorem 1.2, with a little tweak.If V is a real representation of C 4 , and W is a subrepresentation, let C d (V, W ) denote the cofiber of the inclusion Gr ), by our chromatic fixed point theorem, Theorem 1.4.Furthermore, C d (V, W ) C 4 will be the cofiber of the inclusion Gr Now we choose V and W . Recall that L 1 and L 2 were the one dimensional real representation of C 4 and R was the two dimensional irreducible.We let

Thus we have kn (C
Conjecture 6.3.Equality holds in Theorem 6.2. As before, this would be implied by a conjectural calculation of the Q n homology of C d (R n ).Conjecture 6.4.Let m = 2 n+1 − ǫ + 2l with ǫ = 0 or 1, and l ≥ 0. Then .
Our various conjectures imply a conjecture about the behavior of the boundary map If these conjectures are true, then the exactness of the Q n -homology long exact sequence would imply that As expected, the right hand side here is zero if m is even, i.e. ǫ = 0.When m is odd, so ǫ = 1, the right hand side is not zero, but can be rearranged as in the following lemma.Lemma 6.5.If m = 2 n+1 − 1 + 2l and l > 0, then Proof.We expand k G n (d, m − 1): We rewrite k G n (d, m) − kC n (d, m): Subtracting our second expression from the first, and dividing by two, proves the lemma.
Thus we can add the following to our conjectures.Conjecture 6.6.If m = 2 n+1 − 1 + 2l and l > 0, then Example 6.7.Suppose that n = 0, so m = 2l + 1. Conjecture 6.4 predicts that In this appendix we present some tables of calculations that support Conjecture 1.7.For larger Grassmannians the main obstacle to testing the conjecture is the sheer volume of calculations needed.To expedite these calculations the authors used the University of Virginia Rivanna High Performance computing system.The white cells are the conjectured values which have not been checked due to computational limitations.The tables are necessarily symmetric in c and d.
k 1 (Gr d (R d+c )) Thus the corollary follows from the proposition.Remark 2.3.If D = C or H, then Gr d (D m ) has a CW structure with m d cells that are all even dimensional, and thus k n (Gr d (D m )) = m d for all n.
dim Q H * (C d (R 2l+1 ); δ 0 (d, 2l + 1) can be viewed as the dimension of the cokernel of the map i * : H * (Gr d (R 2l+1 ); Q) → H * (Gr d (R 2l ); Q), one can check that our conjectures do correspond to the known behavior of i * -it takes Pontryagin classes to Pontryagin classes -together with the computations dim Q H * (Gr d (R m ); Q) = l c if m = 2l + 1 and d = 2c or 2c + 1 2 l−1 c if m = 2l and d = 2c + 1. Appendix A. Tables (Gr d (R d+c )) (Gr d (R d+c ))