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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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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 THE INDEX OF CERTAIN STIEFEL MANIFOLDS SAMIK BASU AND BIKRAMJIT KUNDU Abstract. This paper computes the Fadell-Husseini index of Stiefel manifolds in the case where the group acts via permutations of the orthogonal vectors. The computations are carried out in the case of elementary Abelian p-groups. The results are shown to imply certain generalizations of the Kakutani-Yamabe-Yujobo theorem. 1. Introduction The existence of equivariant maps between spaces with G-action has been an important question in equivariant topology. The Borsuk-Ulam theorem states that there are no maps n n−1 from S to S , which are equivariant for the C2-action on the spheres via the antipodal map. Ruling out equivariant maps involve invariants of G-spaces, and their computation in the relevant examples. One of the most useful invariants from this point of view is the index. Perhaps the first such index for C2-spaces was defined by Yang [20], in the context of Kakutani’s theorem about proving that for functions on spheres, there are orthogonal vectors which are mapped to the same value. Following along these lines, a number of other indices are defined for C2-spaces which are compared in [13, Chapter 5]. For general G-spaces, the Fadell-Husseini index [9] has been widely used, and is defined as the kernel of the map from the cohomology of BG to the cohomology of the Borel construction. l In this paper, we consider Stiefel manifolds of k-tuples of orthogonal vectors : VkR n (real), Wl,k (complex) and Xl,k (quaternionic). The groups Cp act on the Stiefel manifold n n n for k = p via the homomorphism of Cp → Σpn , which is induced by the left action of Cp on itself. We write ∗ n Z/2[µ1, · · · ,µn] if p = 2 H (BCp ; Z/p)= 2 2 (Z/p[u1, · · · , un, v1, · · · , vn]/(u1, · · · , un) if p is odd, where |µj| = 1. |uj| = 1, and |vj| = 2. We obtain the following computations for the index in this case (see Theorems 4.1, 4.6, 4.3, 4.4, 4.7, 4.8, 4.9). (⌈ l ⌉−1)(p−1) l p−1 n n l • For odd primes p, IndexCp (VpR )= hv i, and when n ≥ 2, IndexCp (Vp R ) ⊂ p−1 p−1 hv1 , · · · , vn i is generated by elements of degree divisible by 2(p − 1). For the l l−1 l prime 2, we have IndexC2 (V2R )= hµ i, and if l is odd, indC2 (V2R ) ≥ l − 1. l (⌈ − ⌉−1)(p−1) • For odd primes p, IndexC (W )= hv p 1 i, and when n ≥ 2, IndexCn (W n ) ⊂ arXiv:2103.02500v1 [math.AT] 3 Mar 2021 p l,p p l,p p−1 p−1 hv1 , · · · , vn i is generated by elements of degree divisible by 2(p − 1). For the 2l−2 prime 2, we have IndexC2 (Wl,2)= hµ i, and if l is odd, indC2 (Wl,2) ≥ 2l − 2. 2l 2l (⌈ p−1 ⌉−1)(p−1) 2l (⌈ p−1 ⌉−2)(p−1) • For odd primes p, IndexCp (Xl,p)= hv i if p | ⌈ p−1 ⌉−1and= hv i p−1 p−1 n n otherwise, and when n ≥ 2, IndexCp (Xl,p ) ⊂ hv1 , · · · , vn i is generated by el- ements of degree divisible by 2(p − 1). For the prime 2, we have IndexC2 (Xl,2) = 4l−4 hµ i, and if l is odd, indC2 (Xl,2) ≥ 4l − 4. The computation of the index is usually done in the context of a concrete problem with geometric consequences. In the context of the topological Tverberg problem, index computations played a crucial role (see [15], [17], [6], [8]). For their uses in other geometric 2010 Mathematics Subject Classification. Primary: 55M20, 55M35; Secondary: 52A35, 55N91. Key words and phrases. Existence of equivariant maps, Stiefel manifolds, Fadell-Husseini index, equivari- ant cohomology. 1 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.
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