What is loop multiplication anyhow? (English) Zbl 1405.55001
In general, it is difficult to calculate the algebra structure of \(H_* C(\Omega X)\) from the singular (or cubical) chain complex \(C_* (\Omega X)\) of a space \(X\). Let \(X\) be a simply connected space and let \(C(X)\) be the normalized \(1\)-reduced chains in the singular complex of \(X\). Then \(C(X)\) has a differential graded coalgebra structure with comultiplication \(\Delta : C(X) \to C(X) \otimes C(X)\). The cobar construction \(\Omega C(X)\) gives a differential graded algebra with a total differential called the Adams-Hilton model of \(X\).
To calculate the algebra structure of \(H_* C(\Omega X)\), J. F. Adams and P. J. Hilton [Comment. Math. Helv. 30, 305–330 (1956; Zbl 0071.16403)] invented an algorithm that associates to a space \(X\) a differential graded algebra \(\Omega C(X)\) whose homology is easy to calculate and is isomorphic to \(H_* C(\Omega X)\) as an algebra; that is, \[ H_* C(\Omega X) \cong H_* \Omega C(X) \] as algebras.
The Eilenberg-Moore geometric approximation theorem [S. Eilenberg and J. C. Moore, ibid. 40, 199–236 (1966; Zbl 0148.43203)] says that there is a natural isomorphism of the homology of the pullback with differential Cotor; that is, \[ H(Y \times_X Z) \cong \text{Cotor}^{C(X)} (C(Y), C(Z)). \]
In this paper, the author describes the natural homology isomorphism \[ HW \cong \text{Cotor}^{CA_1,CA_2,\ldots ,CA_{n-1}}(CX_1, CX_2,\ldots,CX_n) \] as the extension [J. Neisendorfer, Algebraic methods in unstable homotopy theory. Cambridge: Cambridge University Press (2010; Zbl 1190.55001)] of the Eilenberg-Moore geometric approximation theorem to multiple homotopy pullbacks \(W = X_1 \times_{A_1} X_2 \times_{A_2} X_3 \times_{A_3} \times \ldots \times_{A_{n-1}} X_n\) and use this extension to reinterpret the multiplicativity of the homology equivalence.
The author explicitly describes the coalgebra structures in the homology of loop spaces, commutative algebras, rational loop spaces and Quillen’s model category of commutative simply connected rational differential graded coalgebras, and shows how to use the Eilenberg-Moore geometric approximation theorem to compute the mod \(p\) homology of the double loop space of a sphere based on the Bott-Samelson theorem. For example, the author shows that if \(p\) is an odd prime and \(T_p[x]\) is the coalgebra \(\langle 1,x,x^2,\ldots,x^{p-1}\rangle\) with \(x\) of even degree \(2n\), then \[ \text{Cotor}^{T_p[x]} (\mathbb{Z}/p \mathbb{Z} , \mathbb{Z}/p \mathbb{Z}) = E(s^{-1}x) \otimes P(z), \] where deg\((s^{-1}x) = 2n-1\) and deg\((z) = 2pn-2\), and if \(p=2\) and \(x\) has arbitrary degree \(m\), then \[ \text{Cotor}^{T_p[x]} (\mathbb{Z}/2 \mathbb{Z} , \mathbb{Z}/2 \mathbb{Z}) = P(s^{-1}x) \] as primitively generated Hopf algebras, where deg\((s^{-1}x) = m-1\). There are too many beautiful results to be quoted here about loop multiplications.
To calculate the algebra structure of \(H_* C(\Omega X)\), J. F. Adams and P. J. Hilton [Comment. Math. Helv. 30, 305–330 (1956; Zbl 0071.16403)] invented an algorithm that associates to a space \(X\) a differential graded algebra \(\Omega C(X)\) whose homology is easy to calculate and is isomorphic to \(H_* C(\Omega X)\) as an algebra; that is, \[ H_* C(\Omega X) \cong H_* \Omega C(X) \] as algebras.
The Eilenberg-Moore geometric approximation theorem [S. Eilenberg and J. C. Moore, ibid. 40, 199–236 (1966; Zbl 0148.43203)] says that there is a natural isomorphism of the homology of the pullback with differential Cotor; that is, \[ H(Y \times_X Z) \cong \text{Cotor}^{C(X)} (C(Y), C(Z)). \]
In this paper, the author describes the natural homology isomorphism \[ HW \cong \text{Cotor}^{CA_1,CA_2,\ldots ,CA_{n-1}}(CX_1, CX_2,\ldots,CX_n) \] as the extension [J. Neisendorfer, Algebraic methods in unstable homotopy theory. Cambridge: Cambridge University Press (2010; Zbl 1190.55001)] of the Eilenberg-Moore geometric approximation theorem to multiple homotopy pullbacks \(W = X_1 \times_{A_1} X_2 \times_{A_2} X_3 \times_{A_3} \times \ldots \times_{A_{n-1}} X_n\) and use this extension to reinterpret the multiplicativity of the homology equivalence.
The author explicitly describes the coalgebra structures in the homology of loop spaces, commutative algebras, rational loop spaces and Quillen’s model category of commutative simply connected rational differential graded coalgebras, and shows how to use the Eilenberg-Moore geometric approximation theorem to compute the mod \(p\) homology of the double loop space of a sphere based on the Bott-Samelson theorem. For example, the author shows that if \(p\) is an odd prime and \(T_p[x]\) is the coalgebra \(\langle 1,x,x^2,\ldots,x^{p-1}\rangle\) with \(x\) of even degree \(2n\), then \[ \text{Cotor}^{T_p[x]} (\mathbb{Z}/p \mathbb{Z} , \mathbb{Z}/p \mathbb{Z}) = E(s^{-1}x) \otimes P(z), \] where deg\((s^{-1}x) = 2n-1\) and deg\((z) = 2pn-2\), and if \(p=2\) and \(x\) has arbitrary degree \(m\), then \[ \text{Cotor}^{T_p[x]} (\mathbb{Z}/2 \mathbb{Z} , \mathbb{Z}/2 \mathbb{Z}) = P(s^{-1}x) \] as primitively generated Hopf algebras, where deg\((s^{-1}x) = m-1\). There are too many beautiful results to be quoted here about loop multiplications.
Reviewer: Dae-Woong Lee (Jeonju)
MSC:
| 55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |
| 55P35 | Loop spaces |
| 55Q40 | Homotopy groups of spheres |
| 55Q15 | Whitehead products and generalizations |
| 55R05 | Fiber spaces in algebraic topology |
References:
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