THE INDEX of CERTAIN STIEFEL MANIFOLDS 11 Which Is Related to the Pontrjagin Classes of the Associated Bundle V to the Regular Repre- N Sentation Over Cp

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( l ( 1 l, 2. W V V ⌉− 2 p = ) k l,p C n 2)( R R p n n l p ) l − ) ⊂ 1) ⊂ i 2 SAMIK BASU AND BIKRAMJIT KUNDU contexts see [5], [1], [10], [16], [13]. Our index computations yield the following conclusions for geometric problems which are inspired from Kakutani’s Theorem and it’s generalizations (see Corollary 5.5 and Theorems 5.6, 5.7). l−1 m m • Let f : S → R for l ≥ (⌊ 2 ⌋ + 1)(p − 1)+1 for an odd prime p. Then, there are l orthogonal unit vectors v1, · · · , vp in R such that f(v1)= f(v2)= · · · = f(vp). l−1 m m k • Let f : S → R for l ≥ ( 2 + 1)(p − 1)+1 for an odd prime p, then there are k l p orthogonal vectors v1, · · · , vpk in R such that f(v1)= f(v2)= · · · = f(vpk ). l 2l−1 m m • Let f : S(C )= S → R be a map, and suppose that l ≥ (⌊ 2 ⌋ + 1)(p − 1) + 1 l for an odd prime p. Then, there are orthogonal unit vectors v1, · · · , vp in C such that f(v1)= f(v2)= · · · = f(vp). • Let f : S(Hl)= S4l−1 → Rm be a map, and suppose that

1 m m 2 (⌊ 2 ⌋ + 1)(p − 1) + 1 if p | ⌊ 2 ⌋ + 1 l ≥ 1 m m ( 2 (⌊ 2 ⌋ + 2)(p − 1) + 1 if p ∤ ⌊ 2 ⌋ + 1 l for an odd prime p. Then, there are orthogonal unit vectors v1, · · · , vp in H such that f(v1)= f(v2)= · · · = f(vp). (see Theorem 5.7 for more results of the above type)

1.1. Organization. In section 2, we provide the general introduction to the deﬁnitions of the index and the notations used throughout the paper. In section 3, we lay out the method employed in computation of the spectral sequence associated to the homotopy orbits of a Stiefel manifold. In section 4, we carry out the main computations of the index, and ﬁnally, in section 5, we point out the main applications of the computation.

2. Preliminaries In this section, we introduce the basic notations, definitions and results to be used in the paper. We use the category of G-equivariant spaces, and the equivariant index to rule out maps between certain free G-spaces. Throughout G will stand for a finite group. The main reference for the homotopy theoretic results of the section is [14], and the equivariant index and it’s applications is [13]. The category of G-spaces has as objects topological spaces with G-action, and as mor- phisms the G-equivariant maps. An equivariant homotopy between f, g : X → Y is given by a map X × [0, 1] → Y , where G acts trivially on the interval [0, 1]. A G-CW-complex is formed by attaching cells of the type G/H × Dn in increasing dimension. In the category of G-spaces, we write the H-fixed points functor as X 7→ XH for H a subgroup of G. The free G-spaces X satisfy XH = ∅ for all subgroups H which are 6= G. The universal G-bundle EG → BG which classifies principal G-fibrations, satisfies ∗ if H = e EGH ≃ (∅ if H 6= e. One further notes that for a finite G-CW-complex with free action (this is equivalent to the fact that all the cells are of the form G/e × Dn), there is a unique map to EG up to equivariant homotopy. Let V be an orthogonal G-representation. One defines SV to be the one-point compact- ification of V , and uses the orthogonal structure to define G-spaces D(V ) := {v ∈ V : ||v|| ≤ 1} and S(V ) := {v ∈ V : ||v|| = 1}. THEINDEXOFCERTAINSTIEFELMANIFOLDS 3

An easy observation is that S(V )H = S(V H ), which is the sphere in the vector space of H-ﬁxed points of V . We use the following notations for certain important representations to be considered in the paper.

Notation 2.1. We use the notation σ to denote the sign representation of C2. For odd primes p, the notation ξ stands for the 2-dimensional real representation of Cp in which 2π the chosen generator of Cp acts by the rotation of angle p . This is the underlying real representation of a complex 1-dimensional representation which we also denote by ξ. The regular representation of a general group G is denoted by ρG which splits as a direct sum of a trivial 1 dimensional representation and the reduced regular representation which is denoted byρ ¯G. The notation Σn will stand for the symmetric group on n elements. Permuting the n standard basis vectors of R induces an n-dimensional representation of Σn which splits as a direct sum of a trivial 1-dimensional representation generated by the sum of all the vectors and the standard representation which is denoted by W . Recall that as a complex representation the regular representation ρ of a ﬁnite group G is a direct sum of dim(π)-many copies of π, as π varies over the irreducible representations of G. For the group C2, the only irreducible representations are the trivial representation 1 and the sign σ, so that

ρC2 =1+ σ, and this formula is also true as real representations. For the group Cp for an odd prime p, the set of complex irreducible representations are {1,ξ,ξ2, · · · ,ξp−1} where ξr is the tensor product of r copies of ξ, which is the vector space C on which the generator of Cp acts via 2πir multiplication by e p . We thus have 2 p−1 ρCp =1+ ξ + ξ + · · · + ξ as complex representations. As real representations we have p−1 2 2 ρCp =1+ r(ξ)+ r(ξ )+ · · · + r(ξ ), where r stands for the underlying real representation of a complex representation. For n the groups Cp , the set of irreducible representations are given by the tensor products of th irreducible representations of Cp. We write in this case πj as the projection onto the j - ∗ factor, and ξj = πj ξ, and then the regular representation splits as

i in n 1 ρCp = ξ1 ⊗···⊗ ξn . 1≤Xij ≤p Let G be of order n, and consider the action of G on itself by left multiplication. This induces a homomorphism G → Σn. The pullback of the permutation representation of dim n of Σn to G is the regular representation ρ and that of the standard representation is the reduced regular representationρ ¯. A representation V of G may be used to construct a bundle on the classifying space BG : EG ×G V → BG. We often denote this bundle also by the same notation as the representation, and so we often write characteristic classes of representations as elements of H∗(BG). The main results in this paper rely on ruling out equivariant maps between certain G- spaces. For this purpose, we require suitable invariants of equivariant spaces which are computable in simple examples. The index is such an invariant which we recall now. For a G-space X, we denote the orbit space as X/G. However, homology and cohomology computations of the orbit space is not easy. It is easier to make computations for the homotopy orbit space XhG := EG ×G X. For free G-spaces, XhG ≃ X/G. The main technique is the ﬁbration pX X → XhG → BG.

Deﬁnition 2.2. [9] The Fadell-Husseini index IndexG(X) of a G-space X is deﬁned as ∗ ∗ ∗ ∗ Ker(pX ) where pX : H (BG) → H (XhG). 4 SAMIK BASU AND BIKRAMJIT KUNDU

The Fadell-Husseini index is a very useful invariant from the point of view of computations. Some basic properties of index are (see [6])

• Monotonicity: If X → Y is a G-equivariant map, then IndexG(Y ) ⊆ IndexG(X). • Additivity: If (X1 ∪ X2, X1, X2) is an excisive triple of G-spaces, then

IndexG(X1)IndexG(X2) ⊆ IndexG(X1 ∪ X2).

• Join: Let X and Y be G-spaces, then IndexG(X)IndexG(Y ) ⊆ IndexG(X ∗ Y ). The index has an interesting connection to RO(G)-graded equivariant cohomology when G = Cp [2]. n In the case of the elementary Abelian groups Cp , the localization theorem [14, Theorem IV.2.1] has the following consequence for the ideal valued index [11, Cor.1, p. 45]. n Proposition 2.3. Let X be a finite dimensional Cp -CW-complex. The fixed point space G X 6= ∅ ⇐⇒ IndexG(X) 6= 0. n Let us deduce some consequences. A Cp -CW-complex X which is finite dimensional, n such that X is fixed point free, that is XCp = ∅, must have non-trivial ideal valued index. This means that for the fibration n n X → XhCp → BCp , the Serre spectral sequence has a non-trivial differential. From the multiplicative structure ∗ n on the cohomology spectral sequence, the lowest degree class in H (BCp ) must lie in the image of a transgression. This is also the lowest degree class in IndexG(X). Proposition 2.3 allows us to compute the index of representation spheres for the cyclic groups of prime order. In case of the group Cp, a fixed point free space must have free action. The cohomology of BCp may be written as

∗ ∼ Z/2[µ] if p = 2 H (BCp; Z/p) = 2 (Z/p[u, v]/(u ) if p is odd,

Cp Cp with |µ| = 1, |u| = 1, and |v| = 2. As S(V ) = S(V ), IndexCp (S(V )) = 0 ⇐⇒ V contains a copy of the trivial representation. If V Cp = 0, then, dim(V ) is even if p is odd, j as V will be a sum of various ξ . As the fibre of S(V ) → S(V )hCp → BCp is a sphere, there is only one possible non-trivial differential, namely ddim(V ). It follows that hµdim(V )i if p = 2 IndexCp (S(V )) = dim(V ) (hv 2 i if p is odd. In this paper, we are interested in computations of the index for Stiefel manifolds. We fix the notation for this below. l Notation 2.4. The Stiefel manifold, denoted Stk(K ), where K = R, C, or H, is the space of ordered orthonormal k-tuples of vectors in Kl. It is homeomorphic to the quotient space UK(l)/UK(l − k), where UK(l) denotes the unitary group of inner product preserving l transformations. The group Σk acts on Stk(K ) freely by permuting the k orthonormal vectors. l l l l We also write Stk(R ) as Vk(R ), Stk(C ) as Wl,k, and Stk(H ) as Xl,k. The space EG may be filtered by skeleta E(n)G, and this may be used to define a numerical index of G-spaces [13, Chapter 6]. We recall the formulation for G = C2, in n which case the skeleta of EC2 are given by S((n + 1)σ), the space S with antipodal action. The Borsuk-Ulam theorem implies that there are no C2-maps from S((n + 1)σ) → S(nσ), which inspires the following definitions.

Definition 2.5. Let X be a finite C2-CW-complex with free action. Then define

indC2 (X) = min{n ≥ 0 |∃ a C2-map X → S((n + 1)σ)},

coindC2 (X) = max{n ≥ 0 |∃ a C2-map S((n + 1)σ) → X}. THEINDEXOFCERTAINSTIEFELMANIFOLDS 5

We readily conclude that coindC2 (S((n + 1)σ)) = indC2 (S((n + 1)σ)) = n from the

Borsuk-Ulam theorem, and that coindC2 (X) ≤ indC2 (X). The existence C2-map X → Y implies

indC2 (X) ≤ indC2 (Y ), coindC2 (X) ≤ coindC2 (Y ). n+1 From the fact that IndexC2 (S((n + 1)σ)) = hµ i, we see that

coindC2 (X)+1 indC2 (X)+1 hµ i⊂ IndexC2 (X) ⊂ hµ i.

The techniques in [13, Proposition 5.3.2] have the following implications for indC2 and coindC2 .

Proposition 2.6. 1) Suppose that X is a free C2-CW-complex of dimension ≤ n. Then, indC2 (X) ≤ n.

2) Suppose that X is a free C2-space which is r-connected. Then, coindC2 (X) ≥ r + 1.

3. Homotopy orbits of Stiefel manifolds l l Let Stk(K ) be the Stiefel manifold of a k-tuple of orthogonal vectors in K where K l is one of R, C or H. The orthogonal group UK(k) acts on Stk(K ) freely, where UR(k) = O(k), UC(k) = U(k), and UH(k) = Sp(k) the orthogonal, unitary, and symplectic groups respectively. For a subgroup G of UK(k), our goal is to compute the cohomology ring of l l ∗ ∗ l Stk(K )hG ≃ Stk(K )/G, and the map H (BG) → H (Stk(K )hG). l l We note that Stk(K )= UK(l)/UK(l − k), and denote Gk(K )= UK(l)/UK(k) × UK(l − k) as the Grassmannian of k-planes in Kl. Our main technique involves the following diagram of ﬁbrations l l (3.1) Stk(K ) Stk(K )

l / l Stk(K )hG Gk(K )

/ BG BUK(k). We compute the Serre spectral sequence for the left column by using the commutative diagram between spectral sequences and the spectral sequence associated to the right column. The latter spectral sequence is computed via the following commutative diagram of ﬁbrations / l (3.2) UK(l) Stk(K )

l l Gk(K ) Gk(K )

/ B(UK(k) × UK(l − k)) BUK(k).

The top row in the diagram above is given by the quotient map UK(l) → UK(l)/UK(l−k)= l Stk(K ), and the bottom row by the projection onto the ﬁrst factor. In the complex case l Stk(C ) = Wl,k, this has already been computed in [4, Proposition 2.1], which is rewritten below. ∗ Proposition 3.3. The cohomology ring H (Wl,k) is the exterior algebra Λ(yl−k+1, · · · ,yl), l with |yj| = 2j − 1. For the spectral sequence associated to Wl,k → Gk(C ) → BU(k), ′ ′ the classes yj are transgressive with d2j(yj) = −cj, where cj are deﬁned by the equation ′ (1 + c1 + · · · )(1 + c1 + · · · + ck) = 1. 6 SAMIK BASU AND BIKRAMJIT KUNDU

Analogous arguments work for the real and the quaternionic cases. In the quaternionic ∗ case, H (BSp(n)) =∼ Z[q1, · · · ,qn] where the qi in degree 4i are the symplectic Pontrjagin classes. The result for the quaternionic case is described in the Proposition below.

∗ Proposition 3.4. The cohomology H (Xl,k) is the exterior algebra Λ(zl−k+1, · · · , zl), with l |zj| = 4j − 1. For the spectral sequence associated to Xl,k → Gk(H ) → BSp(k), the classes ′ ′ ′ zj are transgressive with d4j(zj ) = −qj, where qj are defined by the equation (1 + q1 + · · · )(1 + q1 + · · · + qk) = 1. In the real case we compute spectral sequences for the fibration l l Vk(R ) → Grk(R ) → BO(k) with Z/2-coefficients, and the fibration l l Vk(R ) → Gr˜ k(R ) → BSO(k) with Z/p-coefficients for odd primes p. In the former case, the result is similar to the Propositions above with the Stiefel-Whitney classes instead of the Chern classes. ∗ l Proposition 3.5. With Z/2-coefficients, the cohomology ring H (Vk(R )) is additively the exterior algebra Λ(ωl−k+1, · · · ,ωl), with |ωj| = j − 1. For the spectral sequence associated l l ′ to Vk(R ) → Gk(R ) → BO(k), the classes ωj are transgressive with dj(ωj) = −wj, where ′ ′ wj are defined by the equation (1 + w1 + · · · )(1 + w1 + · · · + wk) = 1.

l In the latter case, Gr˜ k(R ) is the oriented Grassmannian. We have [7]

ΛZ/p(x1, · · · ,x l−1 ) if l is odd H∗(SO(l); Z/p) =∼ 2 ΛZ/p(x1, · · · ,x l−2 ,ǫl) if l is even ( 2 where |xi| = 4i − 1, |ǫl| = l − 1, and

Z/p[p1, · · · ,p l−1 ] if l is odd H∗(BSO(l); Z/p) =∼ 2 Z/p[p1, · · · ,p l−2 , el] if l is even. ( 2 One deduces

Λ(x l−k+2 , · · · ,x l−1 ,σl−k) if k is odd, l is odd 2 2 Λ(x l−k+1 , · · · ,x l−1 ) if k is even, l is odd ∗ l ∼ 2 2 (3.6) H (VkR ; Z/p) =  Λ(x l−k+1 , · · · ,x l−2 ,ǫl) if k is odd, l is even  2 2 Λ(x l−k+2 , · · · ,x l−2 ,σl−k,ǫl) if k is even, l is even 2 2   where |σj| = j. The spectral sequence computation in cases of interest is described in the Proposition below. l Proposition 3.7. Assume that k is odd. For the spectral sequence associated to Vk(R ) → ˜ l ′ ′ Grk(R ) → BSO(k), the classes xi are transgressive and d4i(xi)= −pi where pi are defined ′ by the equation (1 + p1 + · · · )(1 + p1 + · · · pk) = 1. The classes σl−k (for l odd) and ǫl (for l even) are permanent cycles. Proof. The first part of the proof follows exactly analogously as in [4](Proposition 2.1). We write down the details in the case l is even while the odd case is similar. In the spectral sequence for fibration l SO(l) → Gr˜ kR → B(SO(k) × SO(l − k)), the differentials are d(xi) = pi(γk ⊕ γl−k) [7]. We now use the diagram of fibrations (3.2), l−k noting that for i> 2 , the classes xi pull back to the classes with the same notation under ∗ l q (where q : SO(l) → Vk(R )). We write pi for pi(γk) andp ˜i for pi(γl−k), so that, pi = 0 THEINDEXOFCERTAINSTIEFELMANIFOLDS 7

k l−k if i > 2 andp ˜j = 0 if j > 2 . The formula d(xi) = pi(γk ⊕ γl−k) implies that in the ′ l−k (2(l − k) + 1)-page,p ˜i is equivalent to pi for i ≤ 2 . Thus

d2(l−k+1)(x l−k+1 )= p l−k+1 (γk ⊕ γl−k) 2 2

= pip˜j − i+j= l k+1 X2 ′ = pipj +p ˜l−k+1 2 − i+j= l k+1 ,i≥1 X2 ′ = pipj − i+j= l k+1 ,i≥1 X2 ′ = −p l−k+1 . 2

It now remains to prove that the classes σl−k and ǫl are permanent cycles in the relevant cases. When l − k is even, the class σl−k is the lowest degree non-trivial class which is transgressive for degree reasons. Further it’s degree is even, which forces it to transgress to zero as the Z/p-cohomology groups in BSO(k) are zero in odd degrees. l For l even, it follows by examining the ﬁbration SO(l) → Gr˜ k(R ) → BSO(k)×BSO(l − k) that ǫl is transgressive, and dl(ǫl) is the Euler class of γk ⊕ γl−k. This is zero as k is odd.

l l In order to compute with the Serre spectral sequence Stk(K ) → Stk(K )hG → BG via the map of fibrations (3.1), we note that Propositions 3.3, 3.4, 3.5, 3.7 imply that the differentials are described by values of certain characteristic classes of the universal bundles. These universal bundles are the associated bundle over BUK(k) induced by the action of k l l UK(k) on K . Therefore, the differentials for Stk(K ) → Stk(K )hG → BG are determined by the characteristic classes of the associated G-representation.

4. Index computations l n We compute the index of the Stiefel manifolds Stk(K ) for the groups Cp . The action which we consider are induced by homomorphisms n n Cp → Σpn ⊂ UK(p ), where Cp is included as the cyclic subgroup generated by a p-cycle, and for n ≥ 2, Cpn is included as the transitive action on itself by left multiplication. In either case, note that n n the induced representation on the groups Cp → GLK(p ) is the regular representation over K.

4.1. Computations at odd primes. We start by assuming that p is odd and that the group G = Cp. We also ﬁrst consider the case K = R. The cohomology of BCp is given by ∗ ∼ H (BCp; Z/p) = Z/p[v] ⊗ ΛZ/p[u] where |u| = 1, |v| = 2.

l l m(p−1) Theorem 4.1. Let m = ⌈ p−1 ⌉− 1. Then, IndexCp VpR is the ideal hv i. l l Proof. Consider the ﬁbration VpR → (VpR )hCp → BCp. The associated spectral sequence has E2 page s,t l s t l s+t l E2 (VpR )= H (BCp; H (VpR ; Z/p)) ⇒ H ((VpR )hCp ; Z/p). t l l The Cp-action on H (VpR ; Z/p) is easily observed to be trivial. For, Cp acts on VpR via l the inclusion Cp ⊂ SO(p), and the latter action on VpR as the principal SO(p) bundle 8 SAMIK BASU AND BIKRAMJIT KUNDU

l l VpR → Gr˜ pR . Now as SO(p) is connected, the action of Cp is through maps which are homotopic to the identity. Therefore we may write s,t l s t l s+t l E2 (VpR )= H (BCp) ⊗ H (VpR ) ⇒ H ((VpR )hCp ; Z/p). Recall that the cohomology of the Stiefel manifold is given by (3.6). We note that the universal p-bundle over BSO(p) pulls back to the bundle V over BCp which is associated to the real regular representation of Cp. Therefore, as bundles p−1 2 i V = ⊕i=1 r(ξ ) ⊕ ǫR. We have the following formulae for Pontrjagin classes, p(ξ) = 1 − v2, p(ξi) = (1 − (iv)2)

p−1 2 ⇒ p(V )= (1 − i2v2). Yi=1 k 2k Therefore pk(V ) = (−1) σkv , where σk is the k-th elementary symmetric polynomial in 2 p−1 {i | i = 1, · · · , 2 }. We note that in Z/p this set is precisely the set of quadratic residues, p−1 p−1 p−1 2 and hence are the 2 zeroes of the polynomial x − 1. Hence, σk = 0 for k = 1, · · · , 2 − p 1 p−1 and σ p−1 = 1, and this implies p(V ) = 1 − (−1) 2 v . 2 The diagram (3.1) yields the following commutative diagram

l l VpR VpR

l / ˜ l (VpR )hCp Grp(R )

/ BCp BSO(p). The commutative diagaram allows us to read off the differentials in the spectral sequence for l l (4.2) VpR → (VpR )hCp → BCp from the Serre spectral sequence for l l VpR → Gr˜ p(R ) → BSO(p). The differentials for the associated Serre spectral sequence for the fibration (4.2) are induced by according to Proposition (3.7) ′ d4i(xi)= −pi, dj(ǫl) = 0, and dj(σl−p) = 0.

It follows that in (4.2), xi, ǫl and σl−p are transgressive, and they satisfy ′ d4i(xi)= −pi(V ), dj(ǫl) = 0, and dj(xl−p) = 0, where the classes p′(V ) are defned by the equation p(V )p′(V ) = 1. We have computed that p(V ) = 1 −±vp−1, which implies p′(V )=1+ ±vk(p−1). Xk≥1 Note that whenever l ≥ p and m(p − 1) < l ≤ (m + 1)(p − 1) there is exactly one x i in 2 ∗ l i H (VpR ; Z/p) such that i is divisible by p − 1, and we have p−1 = m. Therefore, the ﬁrst non-trivial diﬀerential in (4.2) is m(p−1) d2m(p−1)(x p−1 )= ±v . m( 2 ) THEINDEXOFCERTAINSTIEFELMANIFOLDS 9

l m(p−1) Hence, IndexCp VpR is the ideal generated by v . Next we consider K = C, and observe that the computation is the same as the real case.

l m(p−1) Theorem 4.3. Let m = ⌈ p−1 ⌉− 1. Then, IndexCp Wl,p is the ideal hv i. Proof. Proceeding analogously as in Theorem 4.1, we observe that the representation induced by the inclusion Cp → U(p) is the regular representation. Therefore, the canonical p-plane bundle pulls back to the complex vector bundle W over BCp induced by the regular representation. The diﬀerentials in the spectral sequence for l l WpC → (WpC )hCp → BCp

′ p−1 i are determined by c (W ). Note that W = ⊕i=1 ξ ⊕ ǫC. We thus have, p−1 i c1(ξ )= iv, c(W )= (1 + iv) Yi=1 p−1 2 − 2 2 p 1 p−1 =⇒ c(W )= (1 − i v ) = 1 − (−1) 2 v Yi=1 =⇒ c′(W )=1+ ±vk(p−1). Xk≥1 m(p−1) The arguments of Theorem 4.1 now imply IndexCp (Wl,p)= hv i. Finally when K = H, the index is described below. Theorem 4.4. Let m be deﬁned by

2l 2l ⌈ p−1 ⌉− 2 if p ∤ ⌈ p−1 ⌉− 1 m = 2l (⌈ p−1 ⌉− 1 otherwise. m(p−1) Then, IndexCp Xl,p is the ideal hv i.

Proof. The quaternionic bundle U over BCp induced by the map Cp → Sp(p) comes from the quaternionic regular representation. As the symplectic Pontrjagin classes are up to sign the even Chern classes of the underlying complex bundle, which for U is two copies of the regular representation, we have − p 1 p−1 2 q(U) = (1 − (−1) 2 v )

− − ′ k(p 1) k(p−1) 2 k(p 1) k(p−1) =⇒ q (U)=(1+ (−1) 2 v ) =1+ (−1) 2 (k + 1)v . Xk≥1 Xk≥1 As in Theorems 4.1 and 4.3, the index is generated by vk(p−1) where k is the smallest integer between 2(l − p + 1) and 2l which is a multiple of (p − 1) and p ∤ k + 1. This is precisely the m described in the statement.

n We next switch out attention to the elementary Abelian groups Cp for n ≥ 2. Our main feature in these computations is to prove certain bounds on the index which turn out to be useful in the examples in the following section. Recall that the group cohomology may be described by ∗ n ∼ H (BCp ; Z/p) = Z/p[v1, · · · , vn] ⊗ ΛZ/p(u1, · · · , un) with |ui| = 1 and |vj| = 2. We prove that the index is contained in the ideal generated by (p − 1)-powers of the vj. We ﬁrst prove the following useful proposition. 10 SAMIKBASUANDBIKRAMJITKUNDU

Proposition 4.5. Suppose Lk is the set of all linear polynomials in Z/p[x1, · · · ,xk]. Then the expression 1 − (1 + l) lY∈Lk p−1 p−1 belongs to the ideal Ip−1(k) := hx1 , · · · ,xk i. Further, it is a sum of monomials in degrees that are multiples of p − 1. Proof. We prove this by induction on k. For k = 1, we have (1 + ax)= a (a−1 + x)= −(xp−1 − 1), × × a∈YZ/p a∈(YZ/p) a∈(YZ/p) which veriﬁes the proposition in this case. Assuming the result for k − 1, we write

(1 + l)= (1 + l) (1 + l + axk). × lY∈Lk l∈YLk−1 a∈(YZ/p) Now we use (z+ax)= (z−ax)= xp (z/x−a)= xp((z/x)p−z/x)= zp−zxp−1 = z(zp−1−xp−1). a∈YZ/p a∈YZ/p a∈YZ/p This allows us to simplify the above as k p−1 p−1 (1 + l + ax )= (1 + l)((1 + l) − xk ). l∈YLk−1 a∈YZ/p l∈YLk−1 Thus, modulo I (k) the expression is equivalent to ( (1 + l))p. By induction p−1 l∈Lk−1 hypothesis, the latter product is congruent to 1 modulo I (k−1). For the second statement p−Q1 we observe that the elementary symmetric polynomials σr in {l ∈ Lk} are non-zero only when r is a multiple of p − 1, in which case they are homogeneous polynomials in the xi of degree a multiple of p − 1. To see this, we proceed as in the above by induction, observing that the result is true for k = 1. For k ≥ 2, we use p−1 p−1 (z − l)= (z − l − axk)= (z − l)((z − l) − xk ). lY∈Lk l∈YLk−1 a∈YZ/p l∈YLk−1 We grade Z/p[x1, · · · ,xk, z] by powers of z, and then note that by induction hypothesis, k−1 the product of z − l over Lk−1 is a sum of elements in gradings of the form p − r(p − 1). p−1 p−1 The product of (z − l) − xk over Lk−1 is symmetric in l ∈ Lk−1 and again by induction hypothesis, this is a expression in the symmetric polynomials of {l ∈ Lk−1}. We now write this as a polynomial over z, noting that the r(p − 1)-degree elementary symmetric s (p−1)t polynomials over Lk−1 occur in expressions σr(p−1)z xk−1 with r(p − 1) + s + t(p − 1) = pk−1(p − 1). It follows that s is divisible by p − 1. As a consequence, the polynomial (z − l) only has terms of degree of the form pk−1 l∈Lk plus a multiple of p − 1, which is also of the form pk plus a multiple of p − 1, as pk − pk−1 Q is divisible by p − 1. It follows that the elementary symmetric polynomials over Lk are non-trivial only in multiples of p−1. The result now follows from the fact that (1+l) l∈Lk is a symmetric polynomial in the l. Q We now return to the index computation of Stiefel manifolds starting with the real case below. p−1 p−1 n n l Theorem 4.6. IndexCp Vp R is contained in the ideal hv1 , · · · , vn i, and is generated by sums of monomials of degree multiples of p − 1 in the vi. Proof. Proceeding as in Theorem 4.1, we compute the spectral sequence associated to the ﬁbration l l n n n n Vp R → (Vp R )hCp → BCp , THE INDEX OF CERTAIN STIEFEL MANIFOLDS 11 which is related to the Pontrjagin classes of the associated bundle V to the regular repre- n sentation over Cp . We thus have

i1 in V = r( ξ1 ⊗···⊗ ξn ) 1≤Xij ≤p ∗ th n where ξj = πj (ξ), the pullback of ξ along the j projection πj : Cp → Cp, and r stands for the underlying real representation. We thus have the following formula on the total Chern classes and Pontrjagin classes,

i1 in i1 in 2 c1(ξ1 ⊗···⊗ ξn )= i1v1 + ... + invn, p(r(ξ1 ⊗···⊗ ξn )) = (1 − (i1v1 + .... + invn) )

2 ⇒ p(V )= (1 − ( ijvj) ) i1Y,..in X Following Proposition (4.5) we see that the Pontrjagin classes of V in positive index are in p−1 p−1 the ideal hv1 , · · · , vn i, and are a sum of homogeneous polynomials of degree a multiple ′ of (p − 1) in the vi. Therefore, the same holds for p (V ). Now Proposition 3.7 and the commutative diagram of ﬁbrations (3.1)

l l Vpn R Vpn R

/ n l n / ˜ n l (Vp R )hCp Grp (R )

n / n BCp BSO(p ) n n n implies the result, as the map Cp → SO(p ) is induces the regular representation on Cp . Although we do not have a precise computation of the index, the result forces that the ﬁrst non-zero class in the index lies in a degree divisible by p − 1. This will be used in the applications in the following section. We have the following analogous result in the complex and the quaternionic case. p−1 p−1 n n n n Theorem 4.7. IndexCp (Wl,p ) and IndexCp (Xl,p ) are contained in the ideal hv1 , · · · , vn i, and are generated by sums of monomials of degree multiples of p − 1 in the vi. Proof. As in Theorem 4.6, we apply Propositions 3.3 and 3.4, and this involves the computation of Chern classes in the complex case, and the symplectic Pontrjagin classes in n the quaternionic case, for the pullback of the universal bundle to BCp . Here W , the pull- n back of the canonical n-dimensional complex bundle to BCp , is induced by the regular representation, and is thus

i1 in W = ξ1 ⊗···⊗ ξn . 1≤Xij ≤p It follows from n i1 in c1(ξ1 ⊗···⊗ ξn )= ijvj Xj=1 that n c(W )= (1 + ijvj) 1≤Yij ≤p Xj=1 which by Proposition 4.5 is a sum of homogeneous polynomials in vj of degree a multiple p−1 p−1 of (p − 1) which also lie in the ideal hv1 , · · · , vn i. Therefore, the same result holds for 12 SAMIKBASUANDBIKRAMJITKUNDU c′(V ), and the result follows. In the quaternionic case, the pullback U of the canonical n bundle to Cp , is twice the regular representation as a complex bundle. It follows that n 2 q(U) = ( (1 + ijvj)) , 1≤Yij ≤p Xj=1 and the rest of the proof follows analogously by Propositions 4.5 and 3.4.

4.2. Computations at p = 2. We follow the same method at the prime 2. The cohomology ∗ ∗ ∞ ring H (BC2; Z/2) = H (RP ; Z/2) = Z/2[µ] where µ is the first Stiefel-Whitney class of canonical line bundle over RP ∞. ∗ Theorem 4.8. As an ideal in H (BC2; Z/2) we have the following computations, l l−1 1) IndexC2 V2R = hµ i. 2(l−1) 2) IndexC2 Wl,2 = hµ i. 4(l−1) 3) IndexC2 Xl,2 = hµ i. Proof. We denote by V (respectively W , U) the real (respectively complex, quaternionic) bundle over BC2 associated to the regular representation. As the regular representation is written as the trivial plus the sign representation, we easily observe the following formulae. w(V )=1+ µ. c(W )=1+ µ2. q(U)=1+ µ4. In the real case, we compute the spectral sequence for the fibration l l V2R → (V2R )hC2 → BC2 using the map of fibrations (3.1)

l l V2R V2R

l / l (V2R )hC2 Gr2(R )

/ BC2 BO(2). Applying Proposition 3.5, the differentials are determined by the formal inverse of w(V )= k ∗ l ∼ 1+ µ which is 1 + k≥1 µ . As H (V2(R ); Z/2) = ΛZ/2(xl−2,xl−1), the first non-zero differential is d (x )= µl−1. This completes the proof in the real case. l−1 lP−2 The complex and the quaternionic cases follow analogously using Propositions 3.3 and 3.4.

Note that if X is a free C2-space which is (r − 1)-connected, then there is a C2-map from S(rσ) → X by C2-equivariant obstruction theory (Proposition 2.6). This implies that r+1 l IndexC2 (X) ⊂ IndexC2 (S(rσ)) = hµ i. Now note that V2(R ) is (l − 3)-connected, so it already follows that the index does not have any classes in degree < l − 1. Theorem 4.8 implies that the condition imposed by the connectivity hypothesis is sharp. This is also true for the complex and quaternionic Stiefel manifolds as Wl,2 is (2l − 4)-connected and Xl,2 is (4l − 6)-connected. n One may now proceed with the computation for the group C2 along the line above. ∗ n ∼ th We have H (BC2 ; Z/2) = Z/2[µ1, · · · ,µn] with |µj| = 1 is the image of µ under the j - projection map. The total Stiefel Whitney class (and also the (mod 2) reduction of the Chern classes and the symplectic Pontrjagin classes) equals the product of all 1+ l ( or a power of this product in case of Chern or symplectic Pontrjagin classes) where l varies THE INDEX OF CERTAIN STIEFEL MANIFOLDS 13 over the linear combinations of the µi. Then, the restriction imposed by Proposition 4.5 is trivial in this case, and we do not obtain any new restrictions on the index other than the ones imposed by the connectivity hypothesis. While the Fadell-Husseini index is not very useful in ruling out equivariant maps in the n case of C2 or C2 , the bound on the C2-index indC2 may be improved by 1 in some cases. We use Steenrod operations to see this.

l Theorem 4.9. Let l be odd. Then, indC2 (V2(R )) ≥ l − 1, indC2 (Wl,2) ≥ 2l − 2, and indC2 (Xl,2) ≥ 4l − 4. l Proof. Applying Theorem 4.8, it remains to rule out C2-maps V2(R ) → S((l −1)σ), Wl,2 → S((2l − 2)σ) and Xl,2 → S((4l − 4)σ). We recall from [18, Ch. IV, §10] that l l−1 l−3 • The l-skeleton of V2(R ) is ≃ to RP /RP . l−1 l−3 • The 2l-skeleton of Wl,2 is ≃ to Σ(CP /CP ). 3 l−1 l−3 • The 4l-skeleton of Xl,2 is ≃ to Σ (HP /HP ). l Suppose that there is a C2-map f : V2(R ) → S((l − 1)σ). This induces a commutative diagram of ﬁbrations

l f / l−2 V2(R ) S

f l hC2 / V2(R )hC2 S((l − 1)σ)hC2

BC2 BC2. This induces a map of spectral sequences. The spectral sequence of the right column l−1 is determined by the differential dl−1(ǫl−2) = µ . From Theorem 4.8 we also have that l−1 dl−1(ωl−2)= µ in the spectral sequence of the left column. In both the spectral sequences, ∗ this is the first non-trivial differential. It follows that f (ǫl−2)= ωl−2. On the other hand as l is odd, so is l − 2, which implies that the degree l − 2 class in ∗ l−1 1 ∗ l H (RP ; Z/2) is mapped by Sq to the degree l−1 class. It follows that in H (V2(R ); Z/2), 1 Sq (ωl−2)= ωl−1. As a consequence we have ∗ 1 f Sq (ǫl−2) = 0 while 1 ∗ 1 Sq f (ǫl−2)= Sq (ωl−2)= ωl−1 6= 0, a contradiction. This completes the proof in the real case. The complex and the quaternionic cases are similar to the above by computing the Steenrod operations using the fact that they commute with the suspension. Under the given hypothesis, Sq2 maps the generator of degree 2l − 3 to the generator in degree 2l − 1 in H∗(Σ(CP l−1/CP l−3; Z/2), and Sq4 maps the generator of degree 4l − 5 to the generator of degree 4l − 1 in H∗(Σ3(HP l−1/HP l−3); Z/2). The result follows analogously.

5. Applications In this section, we describe some applications for our index computations. Our applica- tion is oriented towards the Kakutani’s theorem in geometry and it’s generalizations. The Kakutani theorem in geometry states : For any convex body in Rl there is an l-cube such that the convex body is inscribed in the l-cube. This was proved by Kakutani [12] for n = 3, Yamabe and Yujobo [19] for n ≥ 4. It is implied by the following theorem Theorem 5.1. For any map f : Sl−1 → R, there is an orthonormal basis of vectors l (v1, · · · , vl) of R such that f(vi)= f(vj) for every i and j. 14 SAMIKBASUANDBIKRAMJITKUNDU

The above statement was proved by Yang [20]. We make slightly diﬀerent approach to this, which presents us with a problem in equi- l l variant homotopy theory. From above mentioned f, we construct a map F : VlR → R as F (v1, · · · , vl) = (f(v1), · · · ,f(vl)). l The map F is Σl-equivariant. That is, we have a Σl-action on VlR by

(σ, (v1, · · · , vl)) 7→ (vσ(1), · · · , vσ(l)) l and on R by permuting coordinates. Let W be the standard representation of Σl, so that as a representation the latter is W ⊕ǫ, where ǫ is the trivial one dimensional representation. We l compose F with the projection onto W so that we have a Σl-equivariant map F˜ : VlR → W and we note,

Proposition 5.2. F˜(v1, · · · , vl) = 0 iﬀ f(v1)= f(v2)= · · · = f(vl).

Proof. Note that F˜(v1, · · · , vl) = 0 iff F (v1, · · · , vl) ∈ ǫ. The trivial representation ǫ is embedded as the diagonal in Rl, so the result follows. If l is an odd prime p, a counter-example to Kakutani-Yamabe-Yujobo theorem gives a p Σp equivariant map VpR → W which is non-zero at every point. We restrict to a p-cycle Cp, and also apply the deformation retraction of W − 0 to the unit sphere S(W ). Note that the standard representation W restricts to the reduced regular representationρ ¯ of Cp. This p gives a Cp-map Vp(R ) → S(¯ρ). We compare the Fadell-Husseini indices p−1 2 p p−1 IndexCp (S(¯ρ)) = hv i, and IndexCp (VpR )= hv i p by Theorem 4.1. This rules out any possible Cp-map from VpR → S(W ), implying the n n result in this case. For l = p an odd prime power, we may use the map Cp → Σpn induced n by the action of Cp on itself by left multiplication. In this case we may not deduce the result from the index computations on account of our significantly weaker Theorem 4.6. Finally, if l is not a prime power, restricting to various subgroups of Σl is not useful on account of the existence of maps from EG to S(resG(W )) proved in [3] for many examples of G. In the following we seek a generalization of Kakutani’s theorem replacing the target R by Rm. The results will follow the pattern above : the strongest results for odd primes p, significantly weaker results for prime powers, and no results for non prime powers. The problem we consider here is – Question: Find the integer l(m,n) such that for l ≥ l(m,n) and any map f : Sl → Rm there are n-mutually orthogonal vectors to the same value. l mn Given a f as in the question, we proceed analogously as above defining F : VnR → R by F (v1, · · · , vn) = (f(v1), · · · ,f(vn)). l mn m n We put the Σn action on VnR by permuting the vectors vi, and on R = (R ) by mn m m permuting the coordinates. As a Σn-representation, R decomposes as W ⊕ ǫ , where ǫm includes as the diagonal in (Rm)n. We define F˜ as the projection onto W m and observe

Proposition 5.3. F˜(v1, · · · , vn) = 0 iﬀ f(v1)= f(v2)= · · · = f(vn). Using these notations, we have the following theorem. m Theorem 5.4. 1) Let p be an odd prime. Then l(m,p) ≤ (⌊ 2 ⌋ + 1)(p − 1) + 1. 2) l(m, 2) ≤ m +1 if m is even, and ≤ m +2 if m is odd. Proof. The second statement follows from Theorem 4.9 where it is shown that if l is odd there l is no C2-map from V2R → S((l−1)σ). For the ﬁrst statement, we use the inclusion Cp ⊂ Σp given by the p-cycle (1, · · · ,p), so that W restricts to the reduced regular representation m ρ¯ of Cp. Then, S(¯ρ ) is a free Cp-space whose underlying space is a sphere of dimension m(p−1) m 2 m(p − 1) − 1. It follows that IndexCp (S(¯ρ )) = hv i. THE INDEX OF CERTAIN STIEFEL MANIFOLDS 15

m l m m On the other hand if l ≥ (⌊ 2 ⌋ + 1)(p − 1) + 1, ⌈ p−1 ⌉− 1 is at least (⌊ 2 ⌋ + 1) > 2 . m(p−1) (⌈ l ⌉−1)(p−1) 2 l p−1 Therefore hv i is not contained in IndexCp (VpR )= hv i (by Theorem 4.1). The result follows. Theorem 5.4 may be restated in the following manner for the question stated above. Corollary 5.5. 1) Let f : Sl−1 → Rm for l ≥ m +1 if m is even, and l ≥ m +2 if m is odd. Then there are orthogonal unit vectors v, w in Rl such that f(v)= f(w). l−1 m m 2) Let f : S → R for l ≥ (⌊ 2 ⌋ + 1)(p − 1)+1 for an odd prime p. Then, there are l orthogonal unit vectors v1, · · · , vp in R such that f(v1)= f(v2)= · · · = f(vp). The next result is about prime powers n. l−1 m m k Theorem 5.6. Let f : S → R for l ≥ ( 2 + 1)(p − 1)+1 for an odd prime p, then k l there are p orthogonal vectors v1, · · · , vpk in R such that f(v1) = f(v2) = · · · = f(vpk ). k m k That is, l(m,p ) ≤ ( 2 + 1)(p − 1) + 1. l m Proof. We will rule out the existence of Σpk -maps Vpk R → S(W ). We use the inclusion k k Cp ⊂ Σpk induced by the action of Cp on itself by left multiplication. The standard k representation W of Σpk restricts to the reduced regular representationρ ¯ of Cp . Our l m method involves a comparison of Fadell-Husseini indices of Vpk (R ) and S(W ) via the commutative diagram of ﬁbrations.

l / m Vpk R S(W )

l / m (V k R ) k S(W ) k p hCp hCp

k k BCp BCp

k Since S(W m)Cp = φ, the localization theorem (proposition 2.3) implies that the map ∗ k ∗ m H (BC ; Z/p) → H (S(W ) k ; Z/p) is not injective. Also observe that in the spectral p hCp sequence of the right vertical spectral sequence, the only non-trivial differential is dm(pk−1) which is determined by dm(pk−1)(ǫm(pk−1)−1). The conclusion from the localization theorem m m(pk−1) k implies that this is non-trivial, implying Index k (S(W )) ∩ H (BC ; Z/p) 6= ∅. Cp p l From Theorem 4.6, we observe that the elements of Index k (V k R ) in the lowest degree Cp p is generated by monomials in vi of degree p−1. This is also the image of the first non-trivial l l k differential in the spectral sequence associated to V k R → (V k R ) k → BC . This is both p p hCp p ≥ 2(l − pk + 1) and divisible by 2(p − 1). This implies that the lowest degree terms are in l−pk+1 degree at least 2(⌈ p−1 ⌉)(p − 1). We compute ( m + 1)(pk − 1) + 1 − (pk − 1) m (pk − 1) + 1 2⌈ 2 ⌉(p − 1) = 2⌈ 2 ⌉(p − 1) p − 1 p − 1 m 1 = 2(p − 1)⌈ (1 + p + · · · + pk−1)+ ⌉ 2 p − 1 m k−1 2(p − 1)( 2 (1 + p + · · · + p ) + 1) if mk is even = m k−1 1 (2(p − 1)( 2 (1 + p + · · · + p )+ 2 ) ifmkisodd >m(pk − 1). l m This implies that under the given hypothesis, Index k (V k R ) does not contain Index k S(W ) Cp p Cp k l m and therefore, there cannot be any C -map V k R → S(W ). p Cp 16 SAMIKBASUANDBIKRAMJITKUNDU

We next discuss the complex and quaternionic analogue of these results. For these results, we get the stronger consequence that orthogonal vectors in complex or quaternionic sense are mapped to the same value. We have the following results in this case. Theorem 5.7. A) Let f : S(Cl)= S2l−1 → Rm be a map. Then we have 1) Suppose that 2l ≥ m +2 if m 6≡ 2 (mod 4), and 2l ≥ m +4 for m ≡ 2 (mod 4). Then there are orthogonal unit vectors v, w in Cl such that f(v)= f(w). m 2) Suppose that l ≥ (⌊ 2 ⌋ + 1)(p − 1)+1 for an odd prime p. Then, there are orthogonal l unit vectors v1, · · · , vp in C such that f(v1)= f(v2)= · · · = f(vp). m k k 3) Suppose that l ≥ ( 2 + 1)(p − 1)+1 for an odd prime p. Then, there are p orthogonal l vectors v1, · · · , vpk in C such that f(v1)= f(v2)= · · · = f(vpk ). B) Let f : S(Hl)= S4l−1 → Rm be a map. Then we have 1) Suppose that 4l ≥ m +4 if m 6≡ 4 (mod 8), and 4l ≥ m +8 if m ≡ 4 (mod 8). Then there are orthogonal unit vectors v, w in Hl such that f(v)= f(w). 2) Suppose that 1 m m 2 (⌊ 2 ⌋ + 1)(p − 1) + 1 if p | ⌊ 2 ⌋ + 1 l ≥ 1 m m ( 2 (⌊ 2 ⌋ + 2)(p − 1) + 1 if p ∤ ⌊ 2 ⌋ + 1 l for an odd prime p. Then, there are orthogonal unit vectors v1, · · · , vp in H such that f(v1)= f(v2)= · · · = f(vp). 1 m k k 3) Suppose that l ≥ 2 (( 2 +2)(p −1)+1) for an odd prime p. Then, there are p orthogonal l vectors v1, · · · , vpk in H such that f(v1)= f(v2)= · · · = f(vpk ). Proof. We proceed as in the real case. From f : Sdl−1 → Rm for d = 2 or 4, we may m n m n construct F : Wl,n → (R ) in the case d = 2 and F : Xl,n → (R ) in the case d = 4 by the formula

F (v1, · · · , vn) = (f(v1), · · · ,f(vn)). m n m m The map F is Σn-equivariant, and (R ) splits as a Σn-representation into W ⊕ ǫ where m W is the standard representation of Σn. We deﬁne F˜ to be the projection of F to W , and note that F˜(v1, · · · , vn) = 0 if and only if f(v1)= · · · = f(vn) as in Proposition 5.2. If there is no such tuple of orthogonal vectors, we may normalize the values of F˜ to obtain m m a Σn-equivariant map Wl,n → S(W ) if d =1, or aΣn-equivariant map Xl,n → S(W ). Under the given conditions on l and m we rule out the existence of such maps by restrictions n to subgroups Cp or Cp as in the real case. The statements A1 and B1 follow from the bound on the C2-index in Theorem 4.9. It implies that there is no C2 map Wl,2 → S((2l − 2)σ) and there is no C2-map Xl,2 → S((4l − 4)σ) if l is odd. This implies A1 in the case 4 | m and B1 in the case 8 | m. The remaining statements in A1 follow from the fact that Wl,2 is (2l − 4)-connected and S(mσ) is (m−2)-connected, so that Proposition 2.6 implies the conclusions in the other cases (note that if m is odd, 2l ≥ m + 2 already implies 2l ≥ m + 3). The remaining statements of B1 also follow from Propostion 2.6 using the fact that Xl,2 is (4l − 6)-connected. The statement A2 follows in exactly the same manner as Theorem 5.4 in the p odd case, as the index of the complex Stiefel manifold has the same formula as the real Stiefel manifolds at odd primes (Compare Theorems 4.1 and 4.3). We may deduce B2 from the m index computations in Theorem 4.4 by observing that in the case p | ⌊ 2 ⌋ + 1, 2l (⌊ m ⌋ + 1)(p − 1) + 2 m m ⌈ ⌉− 1 ≥ ⌈ 2 ⌉− 1= ⌊ ⌋ + 1 > , p − 1 p − 1 2 2 m and in the case p ∤ ⌊ 2 ⌋ + 1, 2l (⌊ m ⌋ + 2)(p − 1) + 2 m m ⌈ ⌉− 2 ≥ ⌈ 2 ⌉− 2= ⌊ ⌋ + 1 > , p − 1 p − 1 2 2 and then complete the proof using the index computations for S(¯ρm) in Theorem 5.4. THE INDEX OF CERTAIN STIEFEL MANIFOLDS 17

With respect to the statement A3, we compare the index computation for the complex Stiefel manifold in Theorem 4.7 with the computation for the real Stiefel manifold in The- orem 4.6. We proceed as in Theorem 5.6, to observe that the first non-trivial differential in the Serre spectral sequence associated to the fibration k W k → (W k ) k → BC , l,p l,p hCp p has image in degree ≥ 2(l − pk + 1) and divisible by 2(p − 1). In this case the result follows in an analogous manner as in Theorem 5.6. Observe now that for the fibration k X k → (X k ) k → BC , l,p l,p hCp p Theorem 4.7 implies that the image of the first non-trivial differential is in degree which is both ≥ 4(l − pk + 1) and divisible by 2(p − 1). This implies that the lowest degree terms 2(l−pk+1) are in degree at least 2(⌈ p−1 ⌉)(p − 1). The computation at the end of the proof of m k Theorem 5.6 implies that under the given condition this is > 2 (p − 1). This implies that m Index k S(W ) 6⊂ Index k X k and thus, completes the proof of B3. Cp Cp l,p References [1] D. Baralic,´ P. V. M. Blagojevic,´ R. Karasev, and A. Vuciˇ c´, Index of Grassmann manifolds and orthogonal shadows, Forum Math., 30 (2018), pp. 1539–1572. [2] S. Basu and S. Ghosh, Bredon cohomology of finite dimensional Cp-spaces, to appear in Homology Homotopy Appl. [3] , Equivariant maps related to the topological Tverberg conjecture, Homology Homotopy Appl., 19 (2017), pp. 155–170. [4] S. Basu and B. Subhash, Topology of certain quotient spaces of Stiefel manifolds, Canad. Math. Bull., 60 (2017), pp. 235–245. [5] P. V. M. Blagojevic,´ W. Luck,¨ and G. M. Ziegler, Equivariant topology of configuration spaces, J. Topol., 8 (2015), pp. 414–456. [6] P. V. M. Blagojevic´ and G. M. Ziegler, Beyond the Borsuk-Ulam theorem: the topological Tverberg story, in A journey through discrete mathematics, Springer, Cham, 2017, pp. 273–341. [7] A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2), 57 (1953), pp. 115–207. [8] M. C. Crabb, The topological Tverberg theorem and related topics, J. Fixed Point Theory Appl., 12 (2012), pp. 1–25. [9] E. Fadell and S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk- Ulam and Bourgin-Yang theorems, Ergodic Theory Dynam. Systems, 8∗ (1988), pp. 73–85. [10] Y. Hara, The degree of equivariant maps, Topology Appl., 148 (2005), pp. 113–121. [11] W.-y. Hsiang, Cohomology theory of topological transformation groups, Springer-Verlag, New York- Heidelberg, 1975. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85. [12] S. Kakutani, A proof that there exists a circumscribing cube around any bounded closed convex set in R3, Ann. of Math. (2), 43 (1942), pp. 739–741. [13] J. Matouˇsek, Using the Borsuk-Ulam theorem, Universitext, Springer-Verlag, Berlin, 2003. Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler. [14] J. P. May, Equivariant homotopy and cohomology theory, vol. 91 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. [15] M. Ozaydin, Equivariant maps for the symmetric group, available at https://minds.wisconsin.edu/bitstream/handle/1793/63829/Ozaydin.pdf, (1987). [16] Z. Z. Petrovic´, On equivariant maps between Stiefel manifolds, Publ. Inst. Math. (Beograd) (N.S.), 62(76) (1997), pp. 133–140. [17] A. Y. Volovikov, On the index of G-spaces, Mat. Sb., 191 (2000), pp. 3–22. [18] G. W. Whitehead, Elements of homotopy theory, vol. 61 of Graduate Texts in Mathematics, Springer- Verlag, New York-Berlin, 1978. [19] H. Yamabe and Z. Yujoboˆ, On the continuous function defined on a sphere, Osaka Math. J., 2 (1950), pp. 19–22. [20] C.-T. Yang, On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobôand Dyson. I, Ann. of Math. (2), 60 (1954), pp. 262–282. 18 SAMIKBASUANDBIKRAMJITKUNDU

Stat-Math Unit, Indian Statistical Institute, Kolkata 700108, India. Email address: [email protected], [email protected]

Department of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur Math, Howrah 711202 Email address: [email protected], [email protected]