On the indecomposable elements of the bar construction

Hain, Richard M.

doi:10.1090/S0002-9939-1986-0854039-5

On the indecomposable elements of the bar construction
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by Richard M. Hain PDF

Proc. Amer. Math. Soc. 98 (1986), 312-316 Request permission

Abstract:

An explicit formula for a canonical splitting $s:Q\mathcal {B}({\mathcal {E}^ \cdot }) \to \mathcal {B}({\mathcal {E}^ \cdot })$ of the projection $\mathcal {B}({\mathcal {E}^ \cdot }) \to Q\mathcal {B}({\mathcal {E}^ \cdot })$ of the bar construction on a commutative d.g. algebra ${\mathcal {E}^\cdot }$ onto its indecomposables is given. We prove that $s$ induces a d.g. algebra isomorphism $\Lambda (Q\mathcal {B}({\mathcal {E}^\cdot })) \to \mathcal {B}({\mathcal {E}^\cdot })$ and that $H(Q\mathcal {B}({\mathcal {E}^\cdot }))$ is isomorphic with $QH(\mathcal {B}({\mathcal {E}^\cdot }))$.

References

Kuo Tsai Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), no. 5, 831–879. MR 454968, DOI 10.1090/S0002-9904-1977-14320-6
Richard M. Hain, Iterated integrals and homotopy periods, Mem. Amer. Math. Soc. 47 (1984), no. 291, iv+98. MR 727818, DOI 10.1090/memo/0291

The de Rham homotopy theory of complex algebraic varieties

Walter Michaelis, The dual Poincaré-Birkhoff-Witt theorem, Adv. in Math. 57 (1985), no. 2, 93–162. MR 803009, DOI 10.1016/0001-8708(85)90051-9
John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725