Adams–Hilton model

The Pontryagin algebra $ H _ {*} \Omega X $( cf. also Pontryagin invariant; Pontryagin class) of a topological space $ X $ is an important homotopy invariant (cf. also Homotopy type). It is, in general, quite difficult to calculate the homology $ H _ {*} \Omega X $ directly from the chain complex $ C _ {*} \Omega X $. An algorithm that associates to a space $ X $ a differential graded algebra whose homology is relatively easy to calculate and isomorphic as an algebra to $ H _ {*} \Omega X $ is therefore of great value.

In 1955, J.F. Adams and P.J. Hilton invented such an algorithm for the class of simply-connected CW-complexes [a1]. Presented here in a somewhat more modern incarnation, due to S. Halperin, Y. Félix and J.-C. Thomas [a5], the work of Adams and Hilton can be summarized as follows.

Let $ X $ be a CW-complex such that $ X $ has exactly one $ 0 $- cell and no $ 1 $- cells and such that every attaching mapping is based with respect to the unique $ 0 $- cell of $ X $. There exists a morphism of differential graded algebras inducing an isomorphism on homology (a quasi-isomorphism)

$$ {\theta _ {X} } : {( TV,d ) } \rightarrow {C _ {*} \Omega X } $$

such that $ \theta _ {X} $ restricts to quasi-isomorphisms $ ( TV _ {\leq n } ,d ) \rightarrow C _ {*} \Omega X _ {n + 1 } $, where $ X _ {n + 1 } $ denotes the $ ( n + 1 ) $- skeleton of $ X $, $ TV $ denotes the free (tensor) algebra on a free graded $ \mathbf Z $- module $ V $, and $ \Omega X $ is the space of Moore loops on $ X $. The morphism $ \theta _ {X} $ is called an Adams–Hilton model of $ X $ and satisfies the following properties.

$ ( TV,d ) $ is unique up to isomorphism;

if $ X = * \cup \cup _ {\alpha \in A } e ^ {n _ \alpha + 1 } $, then $ V $ has a degree-homogeneous basis $ \{ {v _ \alpha } : {\alpha \in A } \} $ such that $ { \mathop{\rm deg} } v _ \alpha = n ^ \alpha $;

if $ {f _ \alpha } : {S ^ {n _ \alpha } } \rightarrow {X _ {n _ \alpha } } $ is the attaching mapping of the cell $ e ^ {n _ \alpha + 1 } $, then $ [ \theta ( dv _ \alpha ) ] = {\mathcal K} _ {n _ \alpha } [ f _ \alpha ] $. Here, $ {\mathcal K} _ {n _ \alpha } $ is defined so that

$$ \begin{array}{ccc} \pi _ {n _ \alpha } X _ {n _ \alpha } & \mathop \rightarrow \limits ^ { \approx } &\Omega X _ {n _ \alpha } \\ {} _ { {\mathcal K} _ {n _ \alpha } } \downarrow &{} &\downarrow _ {h} \\ H _ {n _ \alpha - 1 } \Omega X _ {n _ \alpha } & = &H _ {n _ \alpha - 1 } \Omega X _ {n _ \alpha } \\ \end{array} $$

commutes, where $ h $ denotes the Hurewicz homomorphism (cf. Homotopy group).

The Adams–Hilton model has proved to be a powerful tool for calculating the Pontryagin algebra of CW-complexes. Many common spaces have Adams–Hilton models that are relatively simple and thus well-adapted to computations. For example, with respect to its usual CW-decomposition, the Adams–Hilton model of $ \mathbf C P ^ {n} $ is $ ( T ( a _ {1} \dots a _ {n} ) ,d ) $, where $ { \mathop{\rm deg} } a _ {i} = 2i - 1 $ and $ da _ {i} = \sum _ {j + k = i - 1 } a _ {j} a _ {k} $.

Given a cellular mapping $ f : X \rightarrow Y $ between CW-complexes, it is possible to use the Adams–Hilton model to compute the induced homomorphism of Pontryagin algebras. If $ {\theta _ {X} } : {( TV,d ) } \rightarrow {C _ {*} \Omega X } $ and $ {\theta _ {Y} } : {( TW,d ) } \rightarrow {C _ {*} \Omega Y } $ are Adams–Hilton models, then there exists a unique homotopy class of morphisms $ \phi : {( TV,d ) } \rightarrow {( TW,d ) } $ such that $ C _ {*} \Omega g \circ \theta _ {X} $ is homotopic to $ \theta _ {Y} \circ \phi $. Any representative $ \phi $ of this homotopy class can be said to be an Adams–Hilton model of $ g $. In this context, "homotopy" means homotopy in the category of differential graded algebras (see [a2] or [a5] for more details).

One can say, for example, that an Adams–Hilton model of a cellular co-fibration of CW-complexes, i.e., an inclusion of CW-complexes, is a free extension of differential graded algebras. Furthermore, the Adams–Hilton model of the amalgamated sum of an inclusion of CW-complexes, $ \iota $, and any other cellular mapping $ g $ is given by the amalgamated sum of the free extension modelling $ \iota $ and an Adams–Hilton model of $ g $.

Examples of problems to which Adams–Hilton models have been applied to great advantage include the study of the holonomy action in fibrations [a4] and the study of the effect on the Pontryagin algebra of the attachment of a cell to a CW-complex [a3], [a6].

References

[a1]	J.F. Adams, P.J. Hilton, "On the chain algebra of a loop space" Comment. Math. Helv. , 30 (1955) pp. 305–330
[a2]	D.J. Anick, "Hopf algebras up to homotopy" J. Amer. Math. Soc. , 2 (1989) pp. 417–453
[a3]	Y. Félix, J.-M. Lemaire, "On the Pontrjagin algebra of the loops on a space with a cell attached" Internat. J. Math. , 2 (1991)
[a4]	Y. Félix, J.-C. Thomas, "Module d'holonomie d'une fibration" Bull. Soc. Math. France , 113 (1985) pp. 255–258
[a5]	S. Halperin, Y. Félix, J.-C. Thomas, "Rational homotopy theory" , Univ. Toronto (1996) (Preprint)
[a6]	K. Hess, J.-M- Lemaire, "Nice and lazy cell attachments" J. Pure Appl. Algebra , 112 (1996) pp. 29–39