4-Manifolds| 4-Manifolds |
Topology
* 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; Z)
= H2(M2; 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 
and
Hirzebruch signature 
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 R4/(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: R ×
S3 and
R4 admit whole continuous
families of differentiable structures; Compact 4-manifolds may have countably
infinite
sets of them; For example, CP2 #9(–P2).
@ Reviews: Iga IJMPA(02)ht [for
physicists].
@ Invariants: Kronheimer & Mrowka JDG(95);
Fintushel & Stern
JDG(97)
[and manifold operations]; 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 Approach
* Idea: Based on the
study of gauge theories (in particular sself-dual connections); A non-standard
differentiable str on R4 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 R4;
Now, the set
of
all instanton solutions has a differentiable structure, which induces the
nontrivial
one on R4, 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,
= (1/96
2)
Rabcd Rabef 
cdef |g|1/2 d4x .
* Examples: (S4)
= 0; (CP2)
= 1; (S2-bundle
over S2) = 0; (K3)
= 16; (M1 ×
M2) = 0 for any two 2D manifolds M1 and
M2; noncompact, (R2)
= 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].
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];
Taylor
G&T(97)
[and Stein structures]; Mariño & Moore CMP(99)
[non-simply
connected];
Marino ht/00-in
[and topological field theory]; Kronheimer JDG(05)
[from higher-rank bundles]; McDuff BAMS(06)
[and
Floer theory, Ozsváth-Szabó].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
12 jun 2008