By C. R. F. Maunder

Thorough, glossy remedy, primarily from a homotopy theoretic perspective. issues comprise homotopy and simplicial complexes, the elemental staff, homology thought, homotopy thought, homotopy teams and CW-Complexes and different issues. each one bankruptcy comprises workouts and recommendations for extra studying. 1980 corrected variation.

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2. Let y\ < ■ ■ ■ < yn-k and Zk > ■ ■ ■ > z\ be sequences of integers that are complementary in { 1 , . . , n } . Assume that k is even. Then in W we have k l([yi,y2,--- ,yn-k,Zk,zk-i,--- ,zi\) = ^T(n-ZJ). The barred permutations of this type form the poset, denoted W*, of the minimal length left coset representatives of Sn in W. Our terminology and all unexplained notation concerning partitions will follow [Ma]. We set p(k) := (k,k — l,... , 1), a "triangular partition" of length k. DIVIDED DIFFERENCES OF TYPE D 35 Given a strict partition a = (oti > • • • > ati > 0) C p(n — 1), we set a+ := (on + 1, a2 + 1, • • • , « ; + 1) if Z is even, and a+ := (ai + l , a 2 + 1,.

We need one more definition (cf. 3. Since the functions E^,Ek(m) are constant on GV-orbits, we can con sider them as functions on Ay/Gv- The equalities in the following conjec ture are understood in this sense. Conjecture. 1) Up to a sign, £ £ | A J , = Ea, dimV = a £ R+. 2) Up to a sign, 2%(l)|A(, = Ek{\), dim V = S. 3) Let d i m F = m5, m > 1. We conjecture that if x',x" £ Assd are two elements in the same S-equivalence class and Ek(m)(x') ^ 0, E*k{m){x") ^ 0, then E*k(m)(x') = E*k(m){x"). »d as a function on Assd/S-equivalence by setting for an S-equivalence class X: fr*{ \(v\ _ J Ek(m)(x)> 10, */ there exists x £ X with Ek(m){x) otherwise.

Using the braid relations in W one easily shows that if ro £ R(w\), then, after breaking a ribbon in D, we get D' such that rr,/ g R(w\). In the case of the push down operation, it is clear that we get D' with ro — ru1 ■ Note that any configuration of boxes D C Da such that m e R(w\) can be o o obtained from D\ C Du by a sequence of operations of the above described two types. 9. ( "Maximal deformation" of D\ C D^) 42 HAIBAO DUAN AND PIOTR PRAGACZ • Pick the lowest ribbon. Push it down as many times as possible.