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The Pontryagin algebra  $  H _ {*} \Omega X $(
cf. also [[Pontryagin invariant|Pontryagin invariant]]; [[Pontryagin class|Pontryagin class]]) of a [[Topological space|topological space]]  $  X $
is an important [[Homotopy|homotopy]] invariant (cf. also [[Homotopy type|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 [[#References|[a1]]]. Presented here in a somewhat more modern incarnation, due to S. Halperin, Y. Félix and J.-C. Thomas [[#References|[a5]]], the work of Adams and Hilton can be summarized as follows.

Let  $  X $
be a [[CW-complex|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|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 [[#References|[a2]]] or [[#References|[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 [[#References|[a4]]] and the study of the effect on the Pontryagin algebra of the attachment of a cell to a CW-complex [[#References|[a3]]], [[#References|[a6]]].

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