#### 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 __\( \epsilon \)__-isotopy).

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
__\[ B\mathrm{O}(n)\to B\mathrm{PL}(n) \to B\mathrm{TOP}(n) .\]__
A great deal was known
about __\( B\mathrm{O}(n) \)__ but the other two were mysterious.

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
__\[ \mathrm{PL}/\mathrm{O}\to B\mathrm{O}\to B\mathrm{PL} \]__
were understood in terms of the groups of homotopy spheres.

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 __\( \mathbb Z \)__
fibers over a circle if and
only if an obviously necessary finiteness condition holds.
Farrell’s
thesis
[e62]
gives a necessary and sufficient condition
for this to hold in general
in terms of the Whitehead group of the 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 __\( X \)__, __\( S^{\mathrm{PL}}_n(X) \)__.
It consists of simple homotopy equivalences __\( f: M^n \to X \)__ modulo the relation
__\( f_1: M_1^n \to X \)__ is equivalent to __\( f_2: M_2^n \to X \)__ if and only if there
exists a PL homeomorphism
__\( h: M_1 \to M_2 \)__ with __\( f_2\circ h \)__ homotopic to __\( f_1 \)__. There are also
__\( \mathrm{TOP} \)__ and smooth versions and a
version for manifolds with boundary.

For __\( S^{\mathrm{PL}}_n(X) \)__ to be nonempty restricts __\( X \)__: it must satisfy
Poincaré duality since __\( M \)__ does.
Using
Spivak’s
construction
[e51]
of a normal spherical fibration
for Poincaré spaces
and
Atiyah’s
uniqueness result
[e23]
for them, Sullivan
constructed a “differential”
__\[
\mathfrak d: S^{\mathrm{PL}}_n(X)
\to [X,G/\mathrm{PL}]
\]__
provided the Spivak normal fibration,
__\( X\to BG \)__ lifts to __\( B\mathrm{PL} \)__, in analogy with the __\( \mathrm{PL} \)__ to __\( \mathrm{O} \)__
smoothing theory.
Moreover the homotopy fiber of __\( B\mathrm{PL} \to BG \)__ is __\( G/\mathrm{PL} \)__.
From the description of __\( G/\mathrm{PL} \)__ as this homotopy fiber, one can see
it is a homotopy associative, homotopy commutative
H-space.

For a simply connected Poincaré duality space __\( X^n \)__,
Sullivan defined a map
__\[
[X,G/\mathrm{PL}] \xrightarrow{\alpha\,} L_n(\mathbb Z)
\]__
and for __\( n\geqslant 5 \)__,
fit them into the “exact sequence” below. The map __\( \beta \)__ below is just
a version of __\( \alpha \)__ for __\( X\times [0,1] \)__ rel __\( X\times\{0, 1\} \)__.

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 __\( \alpha \)__ and __\( \beta \)__ below to the nonsimply
connected case.
They are defined in all dimensions.
Sullivan’s __\( \mathfrak d \)__ already worked with no __\( \pi_1 \)__ or dimension
assumptions.
The sequence is easiest to explain when __\( X \)__ starts out as a PL manifold __\( M \)__.
The sense in which the sequence is exact needs some explanation.
Here is where __\( n\geqslant 5 \)__ comes in.

The structure set __\( S^{\mathrm{PL}}_n(M) \)__ is only a based set with the
identity as the base point and __\( \alpha \)__ need not be a homomorphism.
Nevertheless, __\( \alpha^{-1}(0) \)__ is the image of __\( \mathfrak d \)__.
The map __\( \scriptstyle\bullet \)__ comes from a group action on a set:
__\[ L_{n+1}(\mathbb Z[\pi_1(M)])\times S^{\mathrm{PL}}_n(M)\to
S^{\mathrm{PL}}_n(M) .\]__
The map __\( \beta \)__ is a homomorphism and the quotient group
__\[ L_{n+1}(\mathbb Z[\pi_1(M)])/\operatorname{image}(\beta) \]__
acts freely on __\( S^{\mathrm{PL}}_n(M) \)__ with orbit space
__\( \alpha^{-1}(0) \)__.
__\begin{equation*}
[\Sigma M, G/\mathrm{PL}] \xrightarrow{\beta\,} L_{n+1}(\mathbb Z[\pi_1(M)])
\xrightarrow{\bullet\,} S^{\mathrm{PL}}_n(M)\xrightarrow{\mathfrak d}
[M,G/\mathrm{PL}]
\xrightarrow{\alpha\,} L_n(\mathbb Z[\pi_1(M)]).
\end{equation*}__

To compute the structure set of a manifold __\( M^n \)__, __\( n\geqslant5 \)__, one needs
to compute two of the four Wall groups.
One also needs to compute the group of normal invariants, __\( [M,G/\mathrm{PL}] \)__
and __\( [\Sigma M, G/\mathrm{PL}] \)__.
Sullivan gave a detailed analysis of the homotopy type of __\( G/\mathrm{PL} \)__
but for Kirby’s needs, only the homotopy
groups are needed. For low dimensions they can be computed from the homotopy
exact sequence of the
fibration and the fact that __\( B\mathrm{O}\to B\mathrm{PL} \)__ is an isomorphism on
homotopy groups for __\( n\leqslant 6 \)__.
For __\( n\geqslant5 \)__ the Sullivan exact sequence computes the groups in
dimensions __\( \geqslant5 \)__
since all homotopy spheres in these dimensions are PL-standard.

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 __\( \mathbb{S}^{n-1} \)__ in __\( \mathbb{S}^n \)__ divides the sphere into two disks, each
with boundary the embedded __\( \mathbb{S}^{n-1} \)__.
Mazur’s proof required an additional hypothesis that the embedding was
“nice” at one point, a condition which was
removed by Morse. Brown gave a self-contained proof of the full result.

The next most complicated result along these lines was to consider
a locally flat embedding of __\( \mathbb{S}^{n-1} \)__ into the interior of an __\( n \)__-disk.
The sphere divides the disk into two pieces: one piece is a smaller __\( n \)__-disk
by the Schoenflies theorem.
The other piece was conjectured to be an annulus, a space homeomorphic
to __\( \mathbb{S}^{n-1}\times[0,1] \)__.
Milnor’s
inclusion of the annulus conjecture on his 1963 problem list from
the Seattle conference
[e42]
plus its inherent simplicity
to state made it an attractive problem which generated some interest.
Cantrell
[e43],
LaBach
[e52]
and even Kirby
[1]
proved versions of the annulus conjecture with “small additional hypothesis”
but
a proof of the full conjecture remained elusive.

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 __\( \mathbb R^n \)__ and
embeddings of __\( \mathbb R^n \)__ in __\( \mathbb R^n \)__.
Kister
[e39]
proved that microbundles are fiber bundles so
that Milnor’s mysterious classifying spaces, __\( B\mathrm{TOP}(n) \)__ are
the classifying spaces of the group of homeomorphisms, __\( \mathbb R^n \to
\mathbb R^n \)__.
These however were also very mysterious. Before 1969 the number of path
components of __\( \operatorname{Homeo}(\mathbb R^n) \)__
was unknown.
Two homeomorphism are in the same path component if and only if they are
isotopic. As a first step, people
tried to determine if
a homeomorphism is isotopic to the identity. Clearly such a homeomorphism
must preserve orientation.

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 __\( \mathbb
R^n \)__ to be one that is a composite of a finite number
of homeomorphisms, each of which is the identity in a neighborhood of
some point.
All orientation-preserving homeomorphisms of __\( \mathbb R^n \)__ which are
differentiable or PL in a neighborhood of a point are stable.
The stable homeomorphism conjecture conjectures that all
orientation-preserving homeomorphisms of __\( \mathbb R^n \)__ are stable.

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 __\( \mathbb R^n \)__ was stable if and only if it was stable
in a neighborhood of any one point.
This gives a notion of stable between different open subsets of __\( \mathbb R^n \)__
and from there to
the notion of a
stable atlas for a manifold and hence the notion of a stable manifold.
There is also the notion of a stable immersion between stable manifolds.

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 __\( n \)__ implies the annulus conjecture in dimension __\( n \)__.

Slightly earlier,
E. Connell
[e31]
proved that stable homeomorphisms
of __\( \mathbb R^n \)__ could be approximated
by PL homeomorphisms if __\( n\geqslant 7 \)__. In the midst of the proof is a
lemma that
bounded homeomorphisms of __\( \mathbb R^n \)__ are stable, __\( n\geqslant 7 \)__.
In 1967,
Bing
[e54]
reduced 7 to 5.

#### 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 __\( \mathbb R^n \)__ are stable, __\( n\geqslant
5 \)__.
The local nature of stable implies that any covering space of a stable
manifold is stable and
if __\( f: N \to M \)__ is a homeomorphism between stable manifolds and if __\( \tilde{f}:
\widetilde{N} \to \widetilde{M} \)__
is the homeomorphism induced between covering spaces, __\( f \)__ is stable if and
only if __\( \tilde{f} \)__ is stable.
Given any homeomorphism __\( f \)__ of __\( \mathbb{T}^n \)__, the induced map on the universal
cover is bounded and
hence stable (__\( n\geqslant 5 \)__) by Connell.
Hence all homeomorphisms __\( \mathbb{T}^n \to \mathbb{T}^n \)__ are stable (__\( n\geqslant 5 \)__).

Kirby says his big breakthrough came one night in August 1968 when he
realized that given a homeomorphism
__\( h: \mathbb R^n \to\mathbb R^n \)__, he could construct a homeomorphism __\( \hat{h} \)__
from a PL manifold, __\( K^n \)__,
to __\( \mathbb{T}^n \)__, __\( n \)__ large.
The homeomorphism __\( h \)__ would be stable if and only if __\( \hat{h} \)__ were stable.

This is not quite enough since __\( \hat{h} \)__ may not be a self-homeomorphism
of __\( \mathbb{T}^n \)__.
If there is a PL homeomorphism __\( g: \mathbb{T}^n\to K^n \)__ then __\( \hat{h} \)__ is stable if
and only if __\( \hat{h}\circ g: \mathbb{T}^n\to \mathbb{T}^n \)__
is stable.

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 __\( \mathbb{T}^n \)__ implies
the stable homeomorphism conjecture
for __\( \mathbb R^n \)__ for __\( n\geqslant 5 \)__.

In the fall of ’68 experts would have probably bet that the Hauptvermutung
for __\( \mathbb{T}^n \)__ was false.
Nevertheless during that fall and the spring of ’69, Kirby, together with
Siebenmann,
tried to make this idea work.
The Hauptvermutung can be attacked via the Sullivan–Wall sequence.
Several phenomena can result in nontrivial elements in __\( S^{\mathrm{PL}}(M^n) \)__.
There might be a pair __\( (N, f) \)__ with __\( N \)__ not homeomorphic to __\( M \)__.
The Hauptvermutung makes no prediction in this case.
There might be __\( (N, f) \)__ with __\( f \)__ a homeomorphism but __\( N \)__ not PL homeomorphic
to __\( M \)__.
In this case the Hauptvermutung is false.
There might be __\( (N, f) \)__ with __\( N \)__ PL homeomorphic to __\( M \)__ but __\( f \)__ not homotopic
to a PL homeomorphism.
The Hauptvermutung still holds in this case.
In general it can be difficult to sort out which elements are which, even
in 2021, but if __\( S^{\mathrm{PL}}_n(M) \)__,
__\( n\geqslant 5 \)__, is a single point the Hauptvermutung holds for __\( M \)__.

Unfortunately, __\( S^{\mathrm{PL}}(\mathbb{T}^n) \)__ is never a single point for __\( n\geqslant 5 \)__
but it can be computed from the Sullivan–Wall sequence.
Fortuitously the calculations of __\( L_k(\mathbb Z[\pi_1(\mathbb{T}^n)]) \)__,
__\( [\mathbb{T}^n, G/\mathrm{PL}] \)__ and __\( [\Sigma \mathbb{T}^n, G/\mathrm{PL}] \)__ can be done
in a similar fashion, which enables an inductive calculation of __\( \alpha \)__
and __\( \beta \)__.
It turns out that __\( \alpha \)__ and __\( \beta \)__ are injective and __\( \operatorname{coker}(\beta) \)__
is naturally isomorphic to
__\( H^3(\mathbb{T}^n;\mathbb Z/2\mathbb Z) \)__, so __\( S^{\mathrm{PL}}(\mathbb{T}^n) \)__ and __\( H^3(\mathbb{T}^n;\mathbb
Z/2\mathbb Z) \)__ have the same number of elements.
The __\( \mathbb Z/2\mathbb Z \)__’s come from the fact that PL surgery problems
over __\( \mathbb{T}^3\times [0,1] \)__ rel __\( \mathbb{T}^3\times\{0,1\} \)__
have signature divisible by 16 by
Rohlin’s
theorem
[e11],
whereas the surgery group realizes all signatures divisible by 8.
Both
Wall
[e56]
and
Hsiang–Shaneson
[e58]
have proofs.

The standard __\( r^n \)__-fold cover of __\( \mathbb{T}^n \)__ is the product of __\( n \)__ copies of the
degree-__\( r \)__ cover on __\( \mathbb{S}^1 \)__.
The explicit form of the identification of __\( S^{\mathrm{PL}}(\mathbb{T}^n) \)__ with __\( H^3(\mathbb{T}^n;\mathbb
Z/2\mathbb Z) \)__ shows that the map
__\( S^{\mathrm{PL}}(\mathbb{T}^n) \to S^{\mathrm{PL}}(\mathbb{T}^n) \)__ induced by the standard __\( r^n \)__-fold cover is
the identity if __\( r \)__ is odd and maps all elements
to the identity homeomorphism if __\( n \)__ is even.

To go back to that night in August, __\( h \)__ is stable if and only if __\( \hat{h} \)__
is stable if and only if __\( \widetilde{\hat{h}} \)__ is stable
where tilde denotes the standard __\( 2^n \)__-fold cover.
But __\( \widetilde{\hat{h}} \)__ is a homeomorphism from a manifold __\( K^n \to \mathbb{T}^n \)__,
where __\( K^n \)__ is PL homeomorphic
to __\( \mathbb{T}^n \)__ and hence __\( \widetilde{\hat{h}} \)__ is stable.

*All orientation-preserving homeomorphisms of* __\( \mathbb R^n \)__ *with*
__\( n\geqslant 5 \)__ *are stable.*

Here is the torus trick.
Start with an orientation-preserving homeomorphism __\( h: \mathbb R^n \to
\mathbb R^n \)__ and pick a PL immersion
of __\( g: \mathbb{T}^n \to \mathbb R^n \)__. The map __\( h\circ g: \mathbb{T}^n \to \mathbb R^n \)__ is a
topological immersion so there is
a, potentially different, PL structure on __\( \mathbb{T}^n \)__, say __\( K^n \)__, such that
__\( h\circ g \)__ is a PL immersion
and hence stable.
There is a map __\( \hat{h}: K^n \to \mathbb{T}^n \)__ which is the identity on the underlying
topological manifolds
and hence a homeomorphism.
Moreover __\( h \)__ is stable if and only if __\( \hat{h} \)__ is stable.

The only problem is that there is no such immersion.
However, the tangent bundle of __\( \mathbb{T}^n \)__ is trivial and if __\( t_0 \)__ is a point in
__\( \mathbb{T}^n \)__, __\( \mathcal T=\mathbb{T}^n - t_0 \)__ is a noncompact manifold.
Hirsch
[e18]
supplies a smooth immersion __\( g: \mathcal T\to \mathbb
R^n \)__.
Let __\( \tau^n \)__ be the smoothing of __\( \mathcal T \)__ for which __\( h\circ g \)__ is smooth
and let __\( h_1: \tau \to \mathcal T \)__
be the induced homeomorphism as above.

There remains the problem of “filling in the hole”.
By Browder–Livesay, __\( n\geqslant 6 \)__, and Wall
[e53]
__\( n=5 \)__, there
is a smooth embedding of
__\( e: \mathbb{S}^{n-1}\times \mathbb R \to \tau \)__ so that
__\( \tau - e(\mathbb{S}^{n-1}\times \mathbb R) \)__ is compact.
Let __\( K=\tau \cup \mathbb R^n \)__, using __\( e \)__ to identify __\( \mathbb R^n -
\vec{0}=\mathbb{S}^{n-1}\times \mathbb R \)__, where __\( \vec{0} \)__ denotes the origin in
__\( \mathbb R^n \)__,
with the image of __\( e \)__ in __\( \tau \)__.
Extend __\( h_1: \tau^n \to \mathcal{T}^n \)__ to __\( \hat{h} \)__ by letting
__\( \hat{h}(\vec{0})=t_0 \)__.
Clearly __\( \hat{h}: K^n \to \mathbb{T}^n \)__ is a bijection.
If __\( U\subset \mathbb{T}^n-t_0 \)__ is open, __\( \hat{h}^{-1}(U)=h_1^{-1}(U) \)__ is open in
__\( \tau \)__ and hence open in __\( K^n \)__.
If __\( U \subset \mathbb{T}^n-t_0 \)__ is closed, it is compact.
Hence __\( \hat{h}^{-1}(U)=h_1^{-1}(U) \)__ is compact in __\( \tau \)__ and hence closed
in __\( K^n \)__.
If __\( U\subset \mathbb{T}^n \)__ is open and __\( t_0\in U \)__,
then __\( \mathbb{T}^n-U \)__ is closed so
__\( \hat{h}^{-1}(\mathbb{T}^n-U)\subset K \)__ is closed and,
since __\( \hat{h} \)__ is a bijection,
__\[ \hat{h}^{-1}(U) = K - \hat{h}^{-1}(X-U) \]__
is open,
so __\( \hat{h} \)__ is continuous and therefore is a homeomorphism.

At some point during academic ’68–69, it was noticed that by crossing
everything with the , __\( n \)__-ball__\( B^n \)__,
the torus trick became a solution to the handle-straightening problem.
After that most of the results needed to put topological manifolds on the
same footing as smooth and PL manifolds
follow.

The most famous was the result that isotopy classes of PL-triangulations
of a topological manifold
__\( M^n \)__, __\( n\geqslant 5 \)__, were given by lifts of the tangent bundle __\[ M
\xrightarrow{\tau_M} B\mathrm{TOP} \]__ to __\( B\mathrm{PL} \)__.
The homotopy groups of __\( \mathrm{TOP}/\mathrm{PL} \)__ were determined.
First it was shown that __\( \mathrm{TOP}/\mathrm{PL} \)__ was either a point or
__\( K(\mathbb Z/2\mathbb Z,3) \)__ and then in the spring,
they showed it was __\( K(\mathbb Z/2\mathbb Z,3) \)__.
It follows from the Serre spectral sequence that there is a class
__\[ \kappa\in H^4(B\mathrm{TOP};\mathbb Z/2\mathbb Z) \]__
such
that a topological manifold __\( M^n \)__, __\( n\geqslant 5 \)__, is PL if and only if
__\( \kappa(M)=\tau_M^\ast(\kappa)=0 \)__.
If __\( M \)__ is PL, define __\( S^{\mathrm{TOP}/\mathrm{PL}}(M) \)__ to be homeomorphisms
__\( f: N \to M \)__ with __\( N \)__ PL modulo the relation
__\( f_1: N_1 \to M \)__ is equivalent to __\( f_2: N_2 \to M \)__ if and only if there
exists a PL homeomorphism
__\( h: N_1\to N_2 \)__ with __\( f_2\circ h \)__ isotopic to __\( f_1 \)__. Then
__\( S^{\mathrm{TOP}/\mathrm{PL}}(M) \)__ and __\( H^3(M;\mathbb Z/2\mathbb Z) \)__ are
bijective.
To see
__\[ \mathrm{TOP}/\mathrm{PL}=K(\mathbb Z/2\mathbb Z,3) ,\]__
they constructed
a nontrivial element in __\( S^{\mathrm{TOP}/\mathrm{PL}}(\mathbb{T}^n) \)__.
For the Arbeitstagung that summer, Siebenmann described a topological
manifold with no
PL structure: see
[e61]
or
page 309 of
[3].

Many long-standing problems with topological manifolds now had solutions.
Since the total space of a normal bundle to __\( M \)__ is PL, compact manifolds
of any dimension have the homotopy type of finite complexes.
This also allows for the assignment of a Whitehead torsion for homotopy
equivalences between manifolds.

Every map __\( f: N \to M \)__ is homotopic to a map __\( g \)__ transverse to a submanifold
__\( P\subset M \)__ with normal microbundle
provided
none of __\( g^{-1}(P) \)__, __\( N \)__ and __\( M \)__ have dimension 4. Manifolds of dimension
__\( \geqslant 6 \)__ have handlebody structures.
Hence surgery and the s-cobordism theorem follow by their usual proofs and
there is a Sullivan–Wall sequence for topological manifolds, __\( n\geqslant5 \)__.
The homotopy groups of the Thom spectra associated to __\( B\mathrm{TOP} \)__
and the oriented version __\( B\mathrm{S}\mathrm{TOP} \)__ give the bordism
groups of topological manifolds and oriented topological manifolds
respectively, except maybe in dimension 4.

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 __\( \geqslant 5 \)__ is now on
an equal footing with differentiable and PL manifolds.

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 __\( E8 \)__ be the result of plumbing eight copies of the cotangent bundle of
__\( \mathbb{S}^2 \)__ according to the Dynkin diagram
for the exceptional Lie group __\( E_8 \)__ and then coning the boundary, which
recall is the Poincaré homology sphere.
Define the Bing-unicorn as the simplicial complex
__\[
E8\times \mathbb{S}^1.
\]__

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 __\( Q \)__-manifolds to prove
the
topological invariance of Whitehead torsion for finite CW complexes
[e67]
and that the
homeomorphism group of a compact __\( Q \)__-manifold is locally contractible
[e65].
An adaptation of Siebenmann’s ideas produced theorems about CE maps between
__\( Q \)__-manifolds
[e68].

The Edwards–West result that compact ANR’s are __\( Q \)__-manifold factors and
Chapman’s work
defines a unique simple-homotopy type for compact ANR’s and homeomorphisms
between ANR’s are simple. See
[e70].

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.