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By Alex Heller, Myles Tierney

ISBN-10: 0123390508

ISBN-13: 9780123390509

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T. Chen, Iterated integrals, fundamental groups and covering spaces, Trans. Amer. Math. Soc. 206 (1975), 8 3 - 9 8 . Κ. T. Chen, Connections, h o l o n o m y and p a t h space homology, Proc. , Amer. Math. Soc. 27 (1975), 39-52. S. Eilenberg a n d S. M a c Lane, O n the g r o u p s of H(U, n), I, Ann. of Math. 58 (1953), 55-106. S. Eilenberg a n d S. M a c Lane, O n the g r o u p s of Η(Π, η), II, Ann. of Math. 60 (1954), 49-139. S. Eilenberg a n d J. C. M o o r e , H o m o l o g y a n d fibrations I, Comment.

Z)(Z, Z ) is isomorphic to the exterior algebra on generators uu u2, . . over Ζ with deg ut= — 1 + deg Vi. This is a known result, which follows from a Koszul resolution. 5. Applications Consider first a differentiable space X with El = £ products in H (A) being zero). , all Massey H(B(A)) « El % B(Jf (Λ)). ]. 1 and, on the cohomological level, Η*{ΩΧ)*ΈΖ(Η{Α9Ζ)), where Τίζ(Η{Α', Z)) denotes the reduced bar construction over Ζ on the integral cohomology ring H(A; Z) a H(A). Example 1. Let w be the normalized volume element of 5", η > 1, M and let A be the D G subalgebra of A(S ) having {1, w} as a basis.

Any C-set M is the coproduct of its indecomposable sub C-sets, and if M, are these indécomposables, then ( y , Mi) are the components of (Y, M). 1. If M is a C-set, then H o m c ( M , ) preserves epimorop phisms if and only if the components of (Y, M ) have right zeros. Moreover H o m € ( M , ) preserves finite coproducts if and only if M is indecomposable, in which case H o m € ( M , ) preserves all coproducts. Hence op H o m c ( M , ) preserves colimits if and only if (Y, M ) has a right zero, or in other words, if and only if M is a retract of a representable.

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Algebra, Topology, and Category Theory. A Collection of Papers in Honor of Samuel Eilenberg by Alex Heller, Myles Tierney


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