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 = \(\displaystyle{1\over96\pi^2}\) ∫ Rabcd Rabef εcdef |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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send feedback and suggestions to bombelli at olemiss.edu – modified 14 jan 2016