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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