THE FINITENESS OBSTRUCTION OF THE HOMOTOPY MIXING OF TWO CW COMPLEXES

We begin with the following definitibn due to G . Mislin . Definition 1 . A CW complex X is of type FP if the singular chain complex of the universal cover of X is Z [ni (X))-chain homotopy equivalent to a complex of finite length 0 a Pn _*n-1 _* , . . . _# PO 4 0 where each P i is a finitely generated and projective Z(n1 (X))-module . If X is of type FP, the finiteness obstruction of C .T .C . Wall is defined by

length 0 a Pn _* P n-1 _* , . . ._# PO 4 0 where each P i is a finitely generated and projective Z(n 1 (X))-module .The fibrewise localization of a fibration X -o X -0 K(n1 (X) ; I shal l denote bv (Z J _x_f ~u, (n .(,v,) : Suppose that X and Y are two spaces with fundamental groups isomorphic to n.I assume that isomorphisms are fixed so the fundamental groups I shall denote simply by n.
Let P U R be a partition of all primes and let h : (X -0 (Y) f 0 be a homotopy equivalence which induces the identity on n.If X and Y are of type FP then Z is also of type FP .
To formulate our next result we shall use some resulta from algebraic K-theory .Consider the following cartesian diagram : sequence : There is an associated Mayer-Vetoris exact Proof follows by combining together two exact sequences from [1) Chapter IX (6 .3)Theorem and using modules which One can show that a p (det (H) ; 1) e im (K 1 (Z/~nl ) a K0 (Z [ n) )) .cone of the following map : Now I give a sketch of proofs .
Let .C(H), be a mapping and let K* (Z) be a mapping cone of The details will appear elsewhere .
If X is of type FP, the finiteness obstruction of C .T .C .Wall is defined by n _ w(X) -E (-1) 1 [P i ) K0 (Z [n 1 (X)]) .i=0 * Whilst finishing this work the author was supported by the University of Oxford Mathematical Prizes Fund and The Royal Society .does not depend on the choice of P * and it is clear that it vanishes if X is finite .If n1 (X) is finitely presented then the converse is also true ([3) Theorem 13) .Now I give a definition of a homotopy mixing I shall use .It will be a sort of a fibrewise homotopy mixinq .Let F «o E -0 B (*) be a fibration with a nilpotent fibre F .A fibration F1 .oE l .oB(**) í.s called a fibrewise P-localization of the fibration (*) if there is a map over B of the fibration (*) into the fibration (**) which on fibres induces the ordinary P-locali.zation(see[2) ch .I .§8 .),

The homotopy pullback of
the following diagram h XP .o()() Of ---o (Y) 0 a YÁ we shall call a fibrewise mixinq of X and Y at P and R. Let us denote by Z the homotopy pullback of this diagram .Then the universal cover Z is the usual Zabrodsky mixinq of X and Y. consider only the case if n is finite .Theorem 1 .

have projective resolutions of length < 1 .Theorem 2 .
The homotopy equivalence H :(XÓ .*(YÓinduces a Q [rr) -chain homotopy equivalence C* (h) : C* (XÓ® Q The singular complexes C* (Xf)® Q and C* (X)® Q are chain homotopy equivalent (resp .C* (0)® Q and C* (Y)(2 Q) .Hence h induces a chain equivalence H : C* (X)r1 Q 4 C* (Y)(2 Q .Ne can suppose that C* (X) tR Q (resp .C* (Y)(2 Q) is a cellular chain complex of X (resp .Y) because the cellular chain complex and the singular one are natural chain homotopy equivalent .Hence if X and Y are finite H is a chain homotopy equivalence between based complexes and its torsion T(H) e K1 (Q[r]) is defined .Now we can state our second result .Suppose that X and Y are finite'complexes with finite fundamental group rr .Then If X and Y are homologically nilpotent i .e .n acts nilpotently on homology of universal covers we can prove much more .The group ring Q[n] splits in the following way nr the corresponding splitting of chain complexes and chain maps .Hence where (C * (X)q Q) n is a Q-modulé and A* (X) is a Q[n] /(7)module .If X and Y are homologically nilpotent then H* (C*(X)(2 Q) and H* (C * (Y),3 Q) are trivial n-modules .Hence complexes A * (X) and A* (Y) are acyclic and the Reidemeister torsions T(X) of A* (X) and r(Y) pf A * (Y) are defined .We w (Z) = ap(T(H)) C * (X) ,2 Q = (C * (X) 9 Q) n e A* (X) and H = H1 Af H2 , have T(H) = (T(H1) ;T(H2)) E K1 (0) x K1 (Q[n)/(E)) .fl det H2i(H) T(H1 ) = i T(H2 ) = T(X) -T(Y) .Hence follows our third theorem .Theorem 3 .If X and Y are homologically nilpotent and finite with finite fundamental group n then w(Z) = aP (det(H),1) + a p (1,T(X) -T(Y)).
z) ® zp e C* (!) ,2 zR -C * (!),» Q .One can show that C(H) * and C* (Z) map into K* (Z) inducing an isomorphism on homology .It follows from (1) that H i ,(C(H)) is a finitely generated Z[TT)-module for every i and HN (C(H) ;M) = 0 for N big enough and M an arbitrary Z[n)-module .Therefore one can find a complex P* of finite length such that every P, i is finitely generated and projective and a map f : P* -* C(H) * inducing an isomorphism on homology .Hence it follows that Z is of type FP .Let d l : C (H) C,, (X) 9 ZP and d 2 : C (H) -0 C * (Y) g Z R be given by d 1 (Y .x 1 .Y 1 ) _ (-1) n-1 x1 .d 2 (Y .x 1 .y 1 ) + (-1) nequivalences .Let s : C(H) * bp Q -0 C * (V)CQ be given by s(y ,x i ,y 1 ) = (_,)ny .Then s is a chain homotopy between Ho(d 1 5ó idQ ) and d2 e id Q* Hence H , (s 1 a id Q ) are also chain homotopic .we can assume that P *,» Zp and P * 12 ZR are free and based .Hence P* 9, Q has two bases and 1 let T(P* (! Q) be the torsion of the identity with respect to these bases .lt follows immediately from the definition of a that ap (T(P * 9 Q)) -w(P* ) = w(Z) .to homotopy we obtain that T(P * 1 Q) = T(s 2 2 id Q ) -1 .T(H) .T(s 1 » id Q ) .Bu t T (s 1 ® id Q ) E K1 (z [n]) and T (a 2 ® id Q ) e K1 (Z R (n)) .Hence a p (T(P * m 4)) = a p (T(H)) and the proofe of Theoreme 1 and 2 are finished .