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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
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