Singular chains and the fundamental group. (English) Zbl 1479.55017
In this paper the authors show that the fundamental group of a path-connected topological space \(X\) is determined by a bialgebra structure of \(\Omega C_*(X)\) and an \(E_{\infty}\) structure of \(C_*(X)\).
Using the Quillen equivalence \(|-|:Set_{\triangle}\rightleftarrows Top:Sing\), the authors work with 0-reduced simplicial sets \(S\). That is \(S\) is a simplicial set with \(S_0\) being a singleton. Let \((C(S),\partial)\) be the normalized simplicial chain complex of \(S\) and let \(\triangle:C(S)\to C(S)\otimes C(S)\) be the Alexander-Whitney coproduct. Then \((C(S),\partial,\triangle)\) is a differential graded (dg) coassociative coalgebra. The cobar construction \(\Omega C(S)=T(s^{-1}C_{>0}(S))\) is the tensor algebra of the desuspension of \(C_{>0}(S)\). However it is known that \(\Omega-\) is not a homotopy invariant. In general a chain map \(f:C\to C'\) is a quasi-isomorphism while \(\Omega (f)\) is not necessarily a quasi-isomorphism. The authors construct a natural chain complex \(C^K(S)\) such that \(C^K(S)\) and \(C(S)\) are quasi-isomorphic as dg coalgebras, \(\Omega C^K(S)\) and \(\Omega C(S)\) are quasi-isomorphic as dg algebras and \(\Omega C^K(f):\Omega C^K(S)\to\Omega C^K(S')\) is a quasi-isomorphism whenever \(f:S\to S'\) is a weak homotopy equivalence. Furthermore they construct a natural coassociative coproduct \(\triangle':\Omega C^K(S)\to\Omega C^K(S)\otimes\Omega C^K(S)\) and turn \(\Omega C^K(S)\) into a dg bialgebra. This bialgebra structure of \(\Omega C^K(S)\) induces a Hopf algebra structure on \(H_0(\Omega C^K(S);k)\cong k[\pi_1(|S|)]\) for a commutative ring \(k\). Let \(Grp\) be a functor which sends a Hopf algebra \(H\) to the group consisting of its group-like elements \(G=\{g\in H\mid \triangle(g)=g\otimes g,\epsilon(g)=1\}\). Then \(Grp(H_0(\Omega C^K(S));k)\) equals the fundamental group of \(|S|\) so the bialgebra \(\Omega C^K(S)\) determines \(\pi(|S|)\).
In addition the authors describe the surjection operad \(\chi\)-algebra structure of \(C^K(S)\), where \(\chi\) is a dg \(E_{\infty}\)-operad. The \(\chi\)-algebra structure determines the bialgebra structure of \(\Omega C^K(S)\) and hence the fundamental group of \(|S|\).
Using the Quillen equivalence \(|-|:Set_{\triangle}\rightleftarrows Top:Sing\), the authors work with 0-reduced simplicial sets \(S\). That is \(S\) is a simplicial set with \(S_0\) being a singleton. Let \((C(S),\partial)\) be the normalized simplicial chain complex of \(S\) and let \(\triangle:C(S)\to C(S)\otimes C(S)\) be the Alexander-Whitney coproduct. Then \((C(S),\partial,\triangle)\) is a differential graded (dg) coassociative coalgebra. The cobar construction \(\Omega C(S)=T(s^{-1}C_{>0}(S))\) is the tensor algebra of the desuspension of \(C_{>0}(S)\). However it is known that \(\Omega-\) is not a homotopy invariant. In general a chain map \(f:C\to C'\) is a quasi-isomorphism while \(\Omega (f)\) is not necessarily a quasi-isomorphism. The authors construct a natural chain complex \(C^K(S)\) such that \(C^K(S)\) and \(C(S)\) are quasi-isomorphic as dg coalgebras, \(\Omega C^K(S)\) and \(\Omega C(S)\) are quasi-isomorphic as dg algebras and \(\Omega C^K(f):\Omega C^K(S)\to\Omega C^K(S')\) is a quasi-isomorphism whenever \(f:S\to S'\) is a weak homotopy equivalence. Furthermore they construct a natural coassociative coproduct \(\triangle':\Omega C^K(S)\to\Omega C^K(S)\otimes\Omega C^K(S)\) and turn \(\Omega C^K(S)\) into a dg bialgebra. This bialgebra structure of \(\Omega C^K(S)\) induces a Hopf algebra structure on \(H_0(\Omega C^K(S);k)\cong k[\pi_1(|S|)]\) for a commutative ring \(k\). Let \(Grp\) be a functor which sends a Hopf algebra \(H\) to the group consisting of its group-like elements \(G=\{g\in H\mid \triangle(g)=g\otimes g,\epsilon(g)=1\}\). Then \(Grp(H_0(\Omega C^K(S));k)\) equals the fundamental group of \(|S|\) so the bialgebra \(\Omega C^K(S)\) determines \(\pi(|S|)\).
In addition the authors describe the surjection operad \(\chi\)-algebra structure of \(C^K(S)\), where \(\chi\) is a dg \(E_{\infty}\)-operad. The \(\chi\)-algebra structure determines the bialgebra structure of \(\Omega C^K(S)\) and hence the fundamental group of \(|S|\).
Reviewer: Tseleung So (Regina)
MSC:
| 55P15 | Classification of homotopy type |
| 55U40 | Topological categories, foundations of homotopy theory |
| 57T30 | Bar and cobar constructions |
| 55P35 | Loop spaces |
| 57M05 | Fundamental group, presentations, free differential calculus |
References:
| [1] | 3. Theorem 2.6: adding the E ∞ -structure. We deduce Theorem 2.6 from constructions in [BeFr04] and observations of [Ka03]. We would like to stress the importance of Kan replacements and using the notion of Ω-quasi-isomorphisms of χ-coalgebras in order to keep the data of the fun-damental group: as a consequence of Theorem 2.6 one may recover both the E ∞ -structure and the fundamental group of S from any χ-coalgebra Ω-quasi-isomorphic to C(K(S)) with the χ-coalgebra structure described in [BeFr04]. In general, this statement is false if we do not perform a Kan References |
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