12 MARTIN ARKOWITZ AND GREGORY LUPTON generated by W4n-1· Define f.. : M ----. M to be the identity on V2n-l and v2n and f..(w4n-d = W4n-1 + AV2n-1V2n, with A =I 0. Then !.Iva =~Iva and thus OI(f.., ~):Vi_____. H*(M) is defined. Note 01(f.., ~)(w4n-1) = [f(w4n-l) -W4n-1] = .X[v2n-1V2n] =I 0, so 01(f..,~) =I 0. But we now show f.. ~~by constructing a homotopy H: M 1 ----. M. Let H start at f.. and set H(ihn) = -~V2n-l· Let H be zero on the other generators. Clearly H is a homomorphism of DG algebras, and by Proposition 4.4, H ends at t. Therefore H is a homotopy from f.. to ~. Thus an additional hypothesis is needed for the converse of Proposition 3.3. We denote by Hom- 1(Vo,H*(N)) the degree -1 homomorphisms of the graded vector space Vo into the graded vector space H* (N). 4.7 Proposition Let M be 2-stage, J,g: M ----. N maps such that !Iva = glva and Hom- 1(Vo,H*(N)) = 0. Iff~ g, then 0 1(f,g) = 0: V1 ----. H*(N). Proof By Corollary 4.5, 01(f,g)(wk) = -[Hidwk], where His a homotopy from f to g. Thus it suffices to show that Hid(wk) is a coboundary. Now d(wk) E A(V 0 ), so d( wk) is a linear combination of terms of the form Vj 1 Vj,, where { v1, ... , Vr} is a basis of V0 and t ~ 2. Hence id(wk) is a linear combination of terms of the form Vj 1 ... fjJk · "Vj,· We show that H(vj 1 .. ·Vjk ... vj,) = H(vjJ .. ·H(vjk) .. ·H(vj,) is a coboundary. First observe that dH(vj) = Hd(vj) = 0 by Lemma 4.3. Thus O(vj) = [H(vj)] defines a homomorphism(): V0 ----. H*(N) of degree -1. By hy- pothesis, () = 0. Therefore, for every j = 1, ... , k, H( Vj) is a co boundary. Hence in the expression H(vjJ · · · H(vjk) · · · H(vj,) the element H(vjk) is a coboundary and the elements H(vj 1 ), ... ,H(vjk_J,H(vjk+J, ... ,H(vj,) are cocycles. Thus H(vjJ · · ·H(vjk) · · ·H(vj,) is a coboundary. This completes the proof. D This result can be used to give conditions under which [M,J\f] is infinite. 4.8 Proposition Let M = A(V 0 , V1 d) be a 2-stage minimal algebra and assume that Hom- 1(V0 , H*(N)) = 0 and that Hom(V1, H*(N)) =I 0. Then the set [M,J\f] is infinite. Proof Since Hom(V1, H* (N)) =I 0, there exist infinitely many distinct homomor- phisms ()i : V1 ----. H*(N). Fix any map f : M ----. Nand apply Lemma 4.1 to obtain maps g : M----. N such that gilva =!Iva and 01(f,gi) = Oi. It suffices to
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