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 M₁ and M₂ are homotopy equivalent iff (i) H₂(M₁; \(\mathbb Z\)) = H₂(M₂; \(\mathbb Z\)), 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 M₁ and M₂ 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 S⁴.
* 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: \(\mathbb R\) × S³ and \(\mathbb R^4\) admit whole continuous families of differentiable structures; Compact 4-manifolds may have countably infinite sets of them; For example, CP² #9(−P²); 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 d⁴x .

* Examples: τ(S⁴) = 0; τ(\(\mathbb C\)P²) = 1; τ(S²-bundle over S²) = 0; τ(K³) = 16; τ(M₁ × M₂) = 0 for any two 2D manifolds M₁ and M₂; 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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