by Laurence R. Taylor
1. Manifolds pre-Kirby
1.1. Smooth and PL manifolds before 1969
Once upon a time, not so long ago, topological manifolds were rather like unicorns. Other kinds of manifolds, like other animals, were plentiful and intensively studied: nonsingular algebraic varieties had been studied for over a century; Poincaré [e77] worked with PL manifolds; and by the early nineteen fifties, many of the fundamental results on smooth manifolds had been developed, including the theorem that smooth manifolds were PL. Still, no one had yet seen either a unicorn or a topological manifold that was not smooth.
Indeed, by the mid-1950s
we knew all topological manifolds of dimension
less than 4
were smooth, where by topological manifold we mean
a locally Euclidean, paracompact, Hausdorff topological space.
As for homeomorphisms, even on the real line there are homeomorphisms that
are not differentiable.
But any homeomorphism between smooth manifolds of dimension less than 4
is isotopic
to a diffeomorphism and the isotopy can be chosen to move every point no
more than a predetermined
positive distance (usually called an
Original citations for these results in dimensions 0 and 1 are unknown to this author. T. Radó [e5] is usually credited with the final details in dimension 2 and Moise [e10] did dimension 3 in a series of papers around 1952.
Milnor’s [e14] 1956 example of a smooth manifold which was homeomorphic but not diffeomorphic to the 7-sphere ended the hope that all homeomorphisms might be isotopic to diffeomorphisms. By 1961 Smale [e25] had proved the h-cobordism theorem, which implied that smooth manifolds homotopy equivalent to a sphere were PL homeomorphic if the dimension was at least 7 (later lowered to 5). This meant that simplicial complexes constructed by Milnor in 1956 were actually PL manifolds with no smooth structure. Other examples followed. Some were by explicit construction but others needed more technique. The Kervaire–Milnor [e29] classification of homotopy spheres in dimensions at least 5 used transversality, surgery theory and the h-cobordism theorem. Handlebody theory was used to classify certain types of smooth manifolds.
Smooth transversality due to Pontryagin [e7] and Thom [e13] works in all dimensions, as does handlebody theory, initiated by Morse [e6] and Bott [e15]. However, other basic tools such as the h-cobordism theorem and surgery theory depend on the Whitney trick and so only were known to work in dimension at least 5.
In 1963 Milnor
[e36]
introduced microbundles for all three kinds
of manifolds and showed manifolds of each
type had tangent bundles of each type. Smooth microbundles are essentially
vector bundles.
There are classifying spaces and maps
Smale [e16] and Hirsch [e18] reduced immersions of smooth manifolds to bundle theory and after microbundles were available Haefliger and Poenaru [e37] produced the analogous results for locally flat PL immersions. Much later, but just before Kirby’s big breakthrough, Lees [e55] proved his immersion theorem which brought topologically locally flat immersion into line with the smooth and PL cases.
During the sixties the theory of PL manifolds caught up with the theory for smooth manifolds. In 1966 Williamson [e46] proved a transversality result for locally flat PL submanifolds. Handlebody theory, the s-cobordism theorem and surgery for PL manifolds in dimensions at least 5 were developed by the combined work of many authors. So now even theorems first proved in the smooth category became available in the PL category with essentially the same proofs.
The questions of when a PL manifold is smooth and when
a PL homeomorphism is isotopic to a diffeomorphism were reduced to lifting
problems for stable bundles
and the homotopy groups of the fibration
The role of algebraic K-theory in the theory of manifolds became clearer.
Problems whose solutions were unobstructed in the simply connected case
became obstructed in the
presence of
a nontrivial fundamental group but necessary and sufficient
conditions for a solution could
be given using algebraic K-theory, coupled with handlebody theory,
transversality and embedding theory.
Smale’s h-cobordism theorem was
extended to the s-cobordism theorem by
Barden
[e32],
Mazur
[e30]
and
Stallings
[e40]
using
J. H. C. Whitehead’s
theory of torsion
[e1].
Wall
[e41]
used the projective class group to study when a
finitely generated, projective chain complex was chain homotopy equivalent
to a finitely generated, free one.
Browder,
Levine
and
Livesay
[e44]
showed that a noncompact manifold with finitely generated homology
which was simply connected at infinity was the interior of a compact manifold
with boundary,
and
Siebenmann’s
thesis
[e45]
gave necessary and sufficient
conditions for this to be true
in general using the projective class group.
Browder and Levine
[e47]
showed that a connected manifold with fundamental
group
Surgery theory as introduced by Wallace [e21], Milnor [e22] and Kervaire–Milnor [e29] is obstructed even in the simply connected case. Browder [e76](originally published in [e26]) and Novikov [e27] introduced the idea of surgery on a normal map.
Classification of compact manifolds in a fixed homotopy type of dimension
at least 5 was becoming a
problem that could be solved.
Sullivan
[e50]
focused attention on the
structure set of a space
For
For a simply connected Poincaré duality space
Wall
[e49]
used quadratic algebraic K-theory, also known as L-theory,
to extend surgery on normal maps to the nonsimply connected case.
Just as in the simply connected case, these groups are 4-fold periodic,
essentially by their construction.
Wall generalized Sullivan’s maps
The structure set
To compute the structure set of a manifold
Of particular relevance to Kirby’s work, Shaneson [e57] and Wall [e60] used Farrell’s thesis to compute the Wall groups of free abelian groups Bass–Heller–Swan [e38] computed the Whitehead, projective class and Nil groups of free abelian groups.
1.2. Topological manifolds before 1969
Alexander [e4] constructed his famous horned sphere in 1924 as an example of an embedding of the 3-ball in the 3-sphere which could not be isotopic to a smooth embedding. In the 1950’s, Bing [e12] and others constructed many strange objects, presciently topological spaces which were not manifolds but which became manifolds after crossing with some Euclidean space.
Results about manifolds without assuming a smooth or PL structure before 1950 were very rare. Algebraic topology results such as Poincaré duality, the Jordon–Brouwer separation theorem and Brouwer’s invariance of domain result were known.
Hanner [e9] proved topological manifolds were ANR’s and hence, by a result of Whitehead’s [e8], the homotopy type of CW complexes. But even compact topological manifolds were not known to always have the homotopy type of finite CW complexes. Wall’s finiteness obstruction [e41] in the projective class group of the fundamental group was known to be defined, but not known to be zero.
Mazur
[e17],
Morse
[e19]
and
Brown
[e20]
proved the topological
Schoenflies theorem in all dimensions which says that a locally flat
embedding of
The next most complicated result along these lines was to consider
a locally flat embedding of
In a positive direction, M. Newman [e48] was able to extend Stallings’s theory of engulfing [e28] to topological manifolds and managed to prove, amongst other things, that any topological manifold homotopy equivalent to a sphere was homeomorphic to a sphere.
In a different, seemingly unrelated direction, people began to study
homeomorphisms of
In the other direction, it is not hard to show that a homeomorphism which
is the identity in a neighborhood of a point
is isotopic to the identity. Define a stable homeomorphism of
Brown and
Gluck
[e33],
[e34],
[e35]
extended these ideas. They defined a
notion of stable in a neighborhood of a point and showed that a
homeomorphism of
It further follows that all orientable smooth and orientable PL manifolds are stable as are all smooth or PL orientation-preserving immersions.
Using this circle of ideas, Brown and Gluck were able to show that the
stable homeomorphism
conjecture in dimension
Slightly earlier,
E. Connell
[e31]
proved that stable homeomorphisms
of
2. Kirby’s breakthrough
Kirby was familiar with Connell’s results since he had used them in his thesis
[2].
Lemma 5 of Connell’s paper
proves that bounded homeomorphisms of
Kirby says his big breakthrough came one night in August 1968 when he
realized that given a homeomorphism
This is not quite enough since
There is an old conjecture, the Hauptvermutung, due to
Steinitz
[e3]
and
Tietze
[e2]
in 1908, which conjectures that if two
simplicial complexes are homeomorphic then they are PL homeomorphic.
Milnor
[e24]
disproved this for complexes but the conjecture was
still open for PL manifolds in 1968.
At this point Kirby had a proof that the Hauptvermutung for
In the fall of ’68 experts would have probably bet that the Hauptvermutung
for
Unfortunately,
The standard
To go back to that night in August,
All orientation-preserving homeomorphisms of
Here is the torus trick.
Start with an orientation-preserving homeomorphism
The only problem is that there is no such immersion.
However, the tangent bundle of
There remains the problem of “filling in the hole”.
By Browder–Livesay,
At some point during academic ’68–69, it was noticed that by crossing
everything with the ,
The most famous was the result that isotopy classes of PL-triangulations
of a topological manifold
Many long-standing problems with topological manifolds now had solutions.
Since the total space of a normal bundle to
Every map
Siebenmann [e59] also pointed out that there was tension between Moise’s 3-manifold results and the high-dimensional results above. A simple example is that 4-dimensional handlebodies are smooth so if there were genuine topological 4-manifolds they could not be handlebodies.
The picture was clarified a decade later when
Freedman
[e74]
proved that Casson handles could be straightened topologically.
Quinn
[e75]
then proved the stable homeomorphism conjecture in dimension 4, so both it
and the annulus conjecture now are known in all dimensions.
The dimension restrictions on transversality were removed and 5-manifolds
acquired handlebody decompositions.
The theory of topological manifolds of dimension
Finally, the author cannot resist showing a genuine topological manifold. Siebenmann and Freedman certainly construct lots of them but they are not easy to visualize the way spheres or projective spaces and such are.
It is said that the pure of heart can sometimes see a unicorn where ordinary
folk see only a horse.
Let
It is a unicorn because it is a very ordinary simplicial complex but only the pure of heart can see that it is actually a manifold [e71].
3. The torus trick post-Kirby
Almost immediately after the introduction of the torus trick, Siebenmann [e63] used a version to prove that a CE map between manifolds of dimension greater than 5 is near a homeomorphism.
A bit later,
T. Chapman
used a handle-straightening theorem for
The Edwards–West result that compact ANR’s are
Daverman [e64] used a theorem of Price and Seebeck [e66] (which also uses handle-straightening) and a torus trick to show that a 1-ULC embedding of manifolds in codimension one is locally flat.
Using some hyperbolic manifolds constructed by Deligne and Sullivan [e69], Sullivan [e72] used covers of these in the same way Kirby used the torus to show that manifolds of dimension at least 5 have Lipschitz and quasiconformal structures.
As late as 2013, M. Hastings [e78] was using the torus trick to classify quantum phases.
By now much of the “trickery” in the torus trick has been subsumed and generalized under the rubric of “controlled surgery”. Quinn’s ends-of-maps papers [e73] are one such sublimation.