By Anthony Bak (auth.), Ib H. Madsen, Robert A. Oliver (eds.)
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Extra info for Algebraic Topology Aarhus 1982: Proceedings of a conference held in Aarhus, Denmark, August 1–7, 1982
Obstructions The unit index of D(e) ~(qp,qq,Dr). is ~(qp,qq,qr ) . is unramified at p,q and r and D(~pqr)/D(np,~q,~ r) has some ramification at p,q and r, the extension D(e)/D(qp,nq,~ r) must have some ramification (and the remark of Q(8p,q), is equal to I, so The maximal real subfield of ~(e) Since ~(~pqr)/~(e) (~) = -I. ~ S(z). where 0o is the ring of integers of Proof: rood 8 and are all zero. To finish the proof of Theorem B we must prove that o4(e) 0x = 0Xo , (2-~p) ( 2 - n q ) ) We choose p ~ 3 mod 4 ; q,r ~ I By Proposition Q(8p,r) (~q , (2-~p)(2-~r) following at p,q and r.
11 ] , Topology, Aarhus 1978, [ 12] Izv. Akad. Nauk. C*-algebra and K-theory, Algebraic Springer-Verlag Lecture Notes in Math. 763. P. 0rlik, Seifert manifolds, Lecture Notes in Math. 291 ~972), Springer-Verlag. [ 13] M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Math. under ihrer ~enzgebiete 68 [ l~] ~972), Springer Verlag. Rees, Novikov's conjecture and manifolds fibering over nonpositively curved manifolds, Thesis, Univ. of Mich. 1980. [ 15] W. Thurston, The geometry and Topology of [ 16] C.
H i c h may have torsion elements) L if Consider the Levi decomposition is the radical of subgroups FS = F ~ S , FL = p (F) As pointed P ~ G be a maximal we have . B. w¥ respectively [ 13] Similarly, are • We have the following facts: \ S/rs (a) So = KS ~) If we set is a compact .. = K \ ~ F S M o and we may give M£ solvamanifold. = __~\L , then we have a fibration 0 a non-positively curved global symmetric space structure. O M~ i s d i f f e o m o r p h i c t o Therefore 0 ~£ and (4) is topologically a trivial fibration.