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A topological space in which each point has a neighbourhood homeomorphic to the four-dimensional Euclidean space  $  \mathbf R  ^ {4} $
or to the closed half-space  $  \mathbf R _ {+}  ^ {4} $. 
This definition is usually extended by the requirement that a four-dimensional manifold be Hausdorff as a topological space, and have a countable base. The topology of four-dimensional manifolds occupies a special place in the [[Topology of manifolds|topology of manifolds]]. On the one hand, the dimension 4 is too small for the straightforward application of techniques of [[General position|general position]]; [[Transversality|transversality]] (and [[Surgery|surgery]]), which are so productive in higher-dimensional topology, yet sufficiently large to exclude the direct use of the more intuitive methods of three-dimensional topology. On the other hand, the topology of four-dimensional manifolds inherits many of the difficulties of three-dimensional and also of higher-dimensional topology. This is illustrated, for example, by the facts that the boundary of a four-dimensional manifold may be an arbitrary three-dimensional manifold, and that every finitely-presented group is the [[Fundamental group|fundamental group]] of some closed four-dimensional manifold. On the latter observation is based the impossibility of algorithmically recognizing homeomorphy of four-dimensional manifolds. The exceptional status of dimension 4 is well-illustrated by the following fact: There exists a non-standard piecewise-linear (and differentiable) structure on the manifold  $  \mathbf R  ^ {n} $
only for  $  n = 4 $. 
There exist four-dimensional manifolds that do not admit piecewise-linear structures. If such a structure exists at all, then there is a unique differentiable structure compatible with it. A four-dimensional manifold equipped with a complex structure is called an analytic surface.

To every closed orientable four-dimensional manifold  $  M $
a unimodular integer-valued symmetric bilinear form  $  L _ {M} $
is associated, acting on the free part of the group  $  H _ {2} ( M ;  \mathbf Z ) $
via the intersections of cycles. The signature of this form is called the signature of the manifold. The intersection form is the most important invariant of a four-dimensional manifold. Two closed simply-connected differentiable four-dimensional manifolds are  $  h $-
cobordant (cf. [[H-cobordism| $  h $-
cobordism]]) if and only if their forms are isomorphic. If the quadratic form corresponding to the bilinear form  $  L _ {M} $
of a differentiable four-dimensional manifold  $  M $
takes only even values, then the structure group SO of its stable tangent bundle can be replaced by the group Spin. Such four-dimensional manifolds are called spin manifolds. There is a topological classification of closed simply-connected four-dimensional manifolds. Every such four-dimensional manifold with an even form is completely determined by it, and every even, integer-valued, symmetric unimodular form is realized as the intersection form of a simply-connected four-dimensional topological manifold. In particular, the four-dimensional topological [[Poincaré conjecture|Poincaré conjecture]] is true. A classifying invariant of four-dimensional manifolds with odd form is given by pairs of the form  $  ( L , \kappa ) $, 
where  $  L $
is an odd, integer-valued, symmetric unimodular bilinear form and  $  \kappa = 0 $
or 1. Every closed simply-connected four-dimensional manifold  $  M $
with an odd form is completely determined by the pair  $  ( L _ {M} , \kappa _ {M} ) $, 
where  $  \kappa _ {M} = 0 $
if the stable tangent bundle of the manifold  $  M $
admits a trivialization, and 1 otherwise. Every such pair is realizable. The question of which forms can be realized by simply-connected differentiable four-dimensional manifolds has not yet been completely answered. It is known that all odd indefinite forms can be realized. Among the even indefinite forms, connected sums of the [[Kummer surface|Kummer surface]] and the manifold  $  S  ^ {2} \times S  ^ {2} $
realize  $  m E _ {8} \oplus n U $, 
if  $  m $
is even and  $  3m \leq  2n $. 
A form of this type with an odd number  $  m $
is known not to be the intersection form of a closed differentiable four-dimensional manifold, since the signatures of such spin manifolds are necessarily divisible by 16, while the signature of the form  $  m E _ {8} \oplus n U $
is equal to  $  8m $. 
From the forms given above, only the forms that are given by unit matrices can be realized.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Mandelbaum,   "Four-dimensional topology: an introduction"  ''Bull. Amer. Math. Soc.'' , '''2'''  (1980)  pp. 1–159</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.H. Freedman,   "The topology of four-dimensional manifolds"  ''J. Differential Geom.'' , '''17'''  (1982)  pp. 357–453</TD></TR></table>

====Comments====
As noted above, the dimension 4 is rather anomalous from several points of view. For piecewise-linear topology that is illustrated by Rokhlin's theorem: Let  $  M  ^ {4} $
be a compact oriented piecewise-linear manifold with second Stiefel–Whitney class  $  w _ {2} ( M  ^ {4} ) = 0 $. 
Then the signature  $  \sigma ( M  ^ {4} ) \equiv 0 $
$   \mathop{\rm mod}  16 $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.K. Donaldson,   "An application of gauge theory to four-dimensional topology"  ''J. Differential Geom.'' , '''18'''  (1983)  pp. 279–315</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.S. Freed,   K.K. Uhlenbeck,   "Instantons and four-manifolds" , Springer  (1984)</TD></TR></table>