By Tammo tom Dieck

ISBN-10: 3540505288

ISBN-13: 9783540505280

This booklet is a jewel– it explains vital, priceless and deep subject matters in Algebraic Topology that you just won`t locate in different places, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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Example text

I. The quadruple (G, B, N, S) is a Tits system. Indeed: Axiom (Tl) follows from Cor. 2 of Prop. 14 of Algebra, Chap. II, § 10, no. 13. Axiom (T2) is proved in Algebra, Chap. I, Correction to p. 97. Axiom (T4) is immediate. e. iB', with B' = wBw- 1 . i + 1 and stabilize the plane spanned by e1 and e1 +1 ; this group is i~omorphic to GL(2, k:). One checks that GjB = BGJ. i)· Identify Gj with GL(2,k); the group BnGj is then identified with the upper triangular subgroup B 2 of GL(2, k), while the group B' n GJ is identified 18 Ch.

S', and put S = {s, s'} and S' = {(ss') 3 ,s',s'(ss') 2 }). 9) Let (W, S) be a Coxeter system with matrix (m(s, s')), and let w+ be the subgroup of W consisting of the elements of even length. Let so E S. Put g 8 = sso. ~ 1 )m(s,s') = 1 for s,s' E S-{so} and m(s,s') =f:. oo, form a presentation of w+. (Let H+ be the group defined by the above presentation. Show that there exists an automorphism a of H+, with square the identity, that transforms g_. to g; 1 for alls E S-{s0 }. ) Show that if the elements of S are conjugate (cf.

From now on we assume that B is a Tits subgroup of G and that we are given a subset S of W satisfying conditions (1), (2) and (3). C(s) = BU C(s) for alls E S. Show that, for alls E Sand all w E W. C(s) C C(w) U C(w"). b) If w E W, the length of w, denoted by l(w), is the smallest integer n ~ 0 for which there exist s 1 •... , Sn ES with C(w) C C(s 1 ) ... C(sn)· Show that l(w) is finite for all w E W. ) < l(v) and lets ES. C(s) (resp. C(u)) (argue by induction on the length of u. C(s) = C(u)UC(v).

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Algebraic Topology and Transformation Groups by Tammo tom Dieck

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