Cobordism independence of Grassmann manifolds
نویسندگان
چکیده
This paper is a continuation of the ongoing study of cobordism of Grassmann manifolds. Let F denote one of the division rings R of reals, C of complex numbers, or H of quaternions. Let t = dimRF . Then the Grassmannian manifold Gk(F) is defined to be the set of all k-dimensional (left) subspaces of Fn+k. Gk(F) is a closed manifold of real dimension nkt. Using the orthogonal complement of a subspace one identifies Gk(F) with Gn(F). In [8], Sankaran has proved that, for F = R,C or H, the Grassmannian manifold Gk(F) bounds if and only if ν(n + k) > ν(k), where, given a positive integer m, ν(m) denotes the largest integer such that 2ν(m) divides m. Given a positive integer d, let G (d) denote the set of bordism classes of all nonbounding Grassmannian manifolds Gk(F) having real dimension d such that k < n. The restriction k < n is imposed because Gk(F)≈ Gn(F) and, for k = n, Gk(F) bounds. Thus, G (d) = {[Gk(F)] ∈ N∗ | nkt = d,k < n, and ν(n + k) ≤ ν(k)} ⊂ Nd . The purpose of this paper is to prove the following:
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