4-Manifolds |

**Topology** > s.a. intersection.

* __Topological classification__:
It is known that a full topological classification is not possible, but one
can make a list "with repetitions" of
all the topologically inequivalent manifolds; Some classes can be completely
classified.

$ __Homotopy equivalence__:
Two simply connected, compact, oriented 4-manifolds *M*_{1}
and *M*_{2} are
homotopy equivalent iff (i) H_{2}(*M*_{1};
\(\mathbb Z\)^{})
= H_{2}(*M*_{2};
\(\mathbb Z\)^{}), and (ii) Their intersection forms are isomorphic.

* __Result__: Two simply
connected compact 4-manifolds may be homotopic, and possibly homeomorphic,
if their Euler number *χ* and Hirzebruch signature *τ* are equal.

$ __Homeomorphism__: Two
simply connected, compact, oriented 4-manifolds *M*_{1} and
*M*_{2} are homeomorphic iff (i) Their
intersection forms are isomorphic, and (ii)
They have the same Kirby-Siebenmann invariant; We thus have a classification with 2 invariants.

* __Decidability__: 1992,
The existence of an algorithmic description of the set of compact 4-manifolds
is an open problem; It depends on the existence
of an algorithm for recognizing S^{4}.

* __Result__: Every connected PL-manifold is
\(\mathbb R\)^{4}/(some homeomorphism group)
[@ Lawrence Top(00)].

**Differentiable Structures**

* __History__: 1982, Approach
to differential topology of 4-manifolds based on gauge theory ideas and Yang-Mills
instantons introduced; 1994, Seiberg-Witten equation introduced as
dual approach; Work by Kronheimer, Mrowka, Morgan, Stern, Taubes.

* __Tasks__: Identify which
topological manifolds are realizable as smooth ones, and classify these up
to diffeomorphisms.

* __Results__: \(\mathbb R\)^{} ×
S^{3} and
\(\mathbb R\)^{}^{4} admit whole continuous
families of differentiable structures; Compact 4-manifolds may have countably
infinite
sets of them; For example, CP^{2} #9(–P^{2});
Gompf constructed uncountably many smoothings of \(\mathbb R\)^{4} which
support Stein structures, Taylor showed that uncountably many smoothings of
\(\mathbb R\)^{}^{4} support no Stein structure.

@ __Reviews__: Iga IJMPA(02)ht [for
physicists].

@ __Invariants__: Kronheimer & Mrowka JDG(95);
Fintushel & Stern
JDG(97)
[and manifold operations]; Taylor
G&T(97)
[and Stein structures]; Pfeiffer PLB(04)gq/03 [and
path integral for quantum gravity]; Fintushel & Stern Top(04)
[different, homeomorphic manifolds with same Seiberg-Witten invariants].

@ __Related topics__: Braungardt & Kotschick Top(05)
[and Einstein metrics]; Akhmedov T&A(04)
[constructing exotic ones].

**Yang-Mills Instanton (Donaldson) Approach**

* __Idea__: Based on the
study of gauge theories (in particular sself-dual connections); A non-standard
differentiable structure on \(\mathbb R\)^{4} can
be obtained from a study of
instanton solutions of some gauge theory; In the limit when the instanton size
goes to zero, the instanton can be associated with a point in \(\mathbb R\)^{}^{4};
Now, the set
of
all instanton solutions has a differentiable structure, which induces the
non-trivial
one on \(\mathbb R\)^{}^{4}, considered as
its boundary.

@ __References__: Donaldson BAMS(83), JDG(83);
Stern MI(83); Fintushel & Stern JDG(84);
Gompf JDG(85); Donaldson Top(90);
Petrie & Randall 90.

**Seiberg-Witten Approach**

* __Idea__: Based on spinor
structures and equations with Dirac operators.

@ __Seiberg-Witten equations__: Seiberg PLB(93), & Witten NPB(94);
Witten JMP(94),
MRL(94)ht;
Donaldson BAMS(96)
[rev].

@ __Generalizations__: Acharya et al NPB(97)
[higher-dimensional]; Park NPB(02)ht/01.

**Other Structures and Concepts** > s.a. 4D
geometry; Surfaces.

* __Hirzebruch signature__: For a compact 4-manifold *M*,

*τ* = \(\displaystyle{1\over96\pi^2}\)
∫ *R*_{abcd}* R*^{ab}_{ef} *ε*^{cdef} |*g*|^{1/2} d^{4}*x* .

* __Examples__: *τ*(S^{4})
= 0; *τ*(\(\mathbb C\)^{}P^{2})
= 1; *τ*(S^{2}-bundle
over S^{2}) = 0; *τ*(*K*^{3})
= 16; *τ*(*M*_{1} ×
*M*_{2}) = 0 for any two 2D manifolds *M*_{1} and
*M*_{2}; Non-compact example, *τ*(\(\mathbb R\)^{}^{2})
= 0.

@ __Subsets__: Morgan & Szabó Top(99) [embedded tori].

@ __Related topics__: Matveyev dg/95,
Akbulut & Matveyev IMRN(98)m.GT/00 [decomposition];
Hegenbarth et al T&A(05)*
[connected sums]; Vajiac IJGMP(10)
[equivariant localization techniques, and relating Donaldson and Seiberg-Witten
invariants].

**References**

@ __Textbooks and reviews__: Lawson 85;
Freedman & Feng 89; Kirby 89;
Donaldson & Kronheimer 90;
Freed & Uhlenbeck 90; Freedman & Quinn 90;
Gompf & Stipsicz 99 [and Kirby calculus].

@ __Homeomorphism__: Mandelbaum BAMS(80);
Brown & Thickstun ed-82; Freedman JDG(83);
van Meter gq/05/CQG [undecidability].

@ __Homotopy equivalence__: Whitehead CMH(49).

@ __Cobordisms__: Kreck G&T(01).

@ __Invariants__: Myers & Periwal NPB(91)
[and topological field theory]; Crane at al ht/93 [Crane-Yetter];
Crane et al ht/94 [details];
Mariño & Moore CMP(99)
[non-simply
connected];
Marino ht/00-conf
[and topological field theory]; Kronheimer JDG(05)
[from higher-rank bundles]; McDuff BAMS(06)
[and
Floer theory, Ozsváth-Szabó].

@ __Related topics__: Chen et al JDG(12) [classification of compact 4-manifolds with positive isotropic curvature].

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