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.
