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Additional resources for Algebra, Algebraic Topology and their Interactions: Proceedings of a Conference held in Stockholm, Aug. 3 - 13, 1983, and later developments

Example text

From the assumptions commutative it follows that the diagrams are and since the lower rows are equal the proof is finished. BSASBSA > ASBSBSA 18]JB81 , ASBSA 1®¢~1 18(~ PA@I , A@A@B , ASB 1818]-1Bt ASBSASB ' t ~pA ~ 1 B@ASB@A ~ B@A@A®B (~ 1 "* B@A@B t~(~®1 ASB@ASB UA~PB//¢ ASASBSB 1~PB> -~ ASB@B PA81~1 t ' / A@B D~ASPB/ AeA@B@B 39 It may be hard in general to check that the conditions of Proposition I are satisfied. The next proposition considers a case when this could be done by a machine (if the algebras involved are finitely presented).

The l e m m a follows if we is a t w o - s i d e d ideal, since a c I and S c I then also k 8 <~> + <8> ~ k c I, which is a r e f o r m u l a t i o n of the claim. First observe that x E I ~ sx E I and xt E I for s E S and t E T , also by (2) we h a v e BS c I , T a c I . Next we c l a i m that it is enough to prove that S c I . Because, T c I and by induction on the length of x C if this is true, then it is p r o v e d that x < a > ~ I . Indeed, suppose x<~> c I , then @(tx) = @ ( t ~ x < ~ > ) ) c <~>k + k < S > < ~ > ~ ~t(<~>k + k<~>)) ~ .

Then equivalence be the space of continuous path at E---,~B is a fibration and y Wy is a fibration. ,, Y~B , I ~ B w(f,y) =f(1); is a continuous map, E B paths factors as the w . ) square y × B E ,, 4 j, and a fibration E c B I × B 1 = {(f,y) If(O) =@y}; the constant wY1 to a continuous we have the B1J-~*E ~---~B of a homotopy struction retract 4 • As with free extensions Existence deformation Y×B E = {(y,e) l~y = we} If (cf. 2 (ii)) , then in the pullback 17 E >B -Y < E '----+B ' +---Y ' is a commutative square of continuous maps topy equivalences in which the vertical and the left arrows are fibrations arrows are weak homo- then YXBE +--Y' XB,E' is a weak homotopy equivalence.

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