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**Sample text**

If M is a monoid and Ζ is projective, then M Proof. The splitting of the augmentation Z M Ζ gives rise to a family of elements e^ ek of M and nonzero coefficients r l 9 . . , rk in Ζ such that Σ r{• = 1 (showing k > 1), and such that Γ Σ ί α ί ? ί = Γ Σ «·*«· for all a e ML Supposing the e{ distinct, we thus see that each a e M induces a permutation π(α) of the ex, and π is a monoid h o m o m o r p h i s m FLATNESS A N D 41 PROJECTIVITY Μ - • S(k) into the symmetric group on k letters. It is easy to see that ZN ->Z splits for every Ν that admits a surjective h o m o m o r p h i s m (functor) M In particular, this applies to the image of π.

The word "differentiable" should be replaced more appropriately by "predifferentia b l e " ) Choose a base point x 0 e X. Then the d e R h a m complex A(X) has an augmentation arising from the base point. Let A be a D G A subalgebra of A{X). Denote by QX the piecewise smooth loop space of I at x 0 . In [3], we have defined a D G subalgebra A' of Λ(ΩΧ) that is spanned by iterated + integrals of the type j" wl · · · w r , w f G A , r > 0. There is a filtration k = Α'{0)α Α'{\)α Α\2)^·· of A' such that ,4'(s) is spanned by iterated integrals of the type J vvx · · · w r , + r < s, WiEA .

It is the direct sum of 2g — 2 free cyclic Z p[ Z ( f ) ] - m o d u l e s . | SOME PROPERTIES OF T W O - D I M E N S I O N A L P O I N C A R É D U A L I T Y G R O U P S 49 We note that this proposition implies one of the results of J. C o h e n [3]. F o r if HX{G; Ζ) = Ζ® Z, then 1 - > K - > G - > Z - > 1 is exact with Κ a free group such that H^K; Zp) ^ Zp. This implies K^Z. T h e only extension of Ζ by Ζ that is a surface group is Ζ © Ζ . 2. Proof of Theorem A Let a: G -» G' be a h o m o m ö r p h i s m of surface groups such that H+(a; Z ) is an isomorphism.