HIGH DIMENSIONAL MANIFOLD TOPOLOGY THEN AND NOW


 Nathaniel Ronald Glenn
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1 1 HIGH DIMENSIONAL MANIFOLD TOPOLOGY THEN AND NOW Andrew Ranicki (Edinburgh) aar Orsay 7,8,9 December 2005 An ndimensional topological manifold M is a paracompact Hausdorff topological space which is locally homeomorphic to R n. Also called a TOP manifold. TOP manifolds with boundary (M, M), locally (R n +, R n 1 ). High dimensional = n 5. Then = before KirbySiebenmann (1970) Now = after KirbySiebenmann (1970)
2 Time scale 1905 Manifold duality (Poincaré) 1944 Embeddings (Whitney) 1952 Transversality, cobordism (Pontrjagin, Thom) 1952 Rochlin s theorem 1953 Signature theorem (Hirzebruch) 1956 Exotic spheres (Milnor) 1960 Generalized Poincaré Conjecture and hcobordism theorem for DIFF, n 5 (Smale) BrowderNovikovSullivanWall surgery theory for DIFF and PL, n Topological invariance of rational Pontrjagin classes (Novikov) 1968 Local contractibility of Homeo(M) (Chernavsky) 1969 Stable Homeomorphism and Annulus Theorems (Kirby) 1970 KirbySiebenmann breakthrough: highdimensional TOP manifolds are just like DIFF and PL manifolds, only more so! 2
3 3 The triangulation of manifolds A triangulation (K, f ) of a space M is a simplicial complex K together with a homeomorphism f : K = M. M is compact if and only if K is finite. A DIFF manifold M can be triangulated, in an essentially unique way (Cairns, Whitehead, 1940). A PL manifold M can be triangulated, by definition. What about TOP manifolds? In general, still unknown.
4 4 Are topological manifolds at least homotopy triangulable? A compact TOP manifold M is an ANR, and so dominated by the compact polyhedron L = K of a finite simplicial complex K, with maps f : M L, g : L M and a homotopy gf 1 : M M (Borsuk, 1933). M has the homotopy type of the noncompact polyhedron ( k= L [k, k + 1] ) /{(x, k) (fg(x), k + 1) x L, k Z} Does every compact TOP manifold M have the homotopy type of a compact polyhedron? Yes (K.S., 1970)
5 5 Are topological manifolds triangulable? Triangulation Conjecture Is every compact ndimensional TOP manifold M triangulable? Yes for n 3 (Moïse, 1951) No for n = 4 (Casson, 1985) Unknown for n 5. Is every compact ndimensional TOP manifold M a finite CW complex? Yes for n 4, since M has a finite handlebody structure (K.S., 1970)
6 Homology manifolds and Poincaré duality 6 A space M is an ndimensional homology manifold if { Z if r = n H r (M, M {x}) = (x M). 0 if r n A compact ANR ndimensional homology manifold M has Poincaré duality isomorphisms [M] : H n (M) = H (M) with [M] H n (M) a fundamental class; twisted coefficients in the nonorientable case. An ndimensional TOP manifold is an ANR homology manifold, and so has Poincaré duality in the compact case. Compact ANR homology manifolds with boundary (M, M) have PoincaréLefschetz duality H n (M, M) = H (M).
7 7 Are topological manifolds combinatorially triangulable? The polyhedron K of a simplicial complex K is an ndimensional homology manifold if and only if the link of every simplex σ K is a homology S (n σ 1). An ndimensional PL manifold is the polyhedron M = K of a simplicial complex K such that the link of every simplex σ K is PL homeomorphic S (n σ 1). A PL manifold is a TOP manifold with a combinatorial triangulation. Combinatorial Triangulation Conjecture Does every compact TOP manifold have a PL manifold structure? No: by the K.S. PLTOP analogue of the classical DIFF PL smoothing theory, and the determination of TOP/PL. There exist noncombinatorial triangulations of any triangulable TOP manifold M n for n 5 (Edwards, Cannon, 1978)
8 8 The Hauptvermutung: are triangulations unique? Hauptvermutung (Steinitz, Tietze, 1908) For finite simplicial complexes K, L is every homeomorphism h : K = L homotopic to a PL homeomorphism? i.e. do K, L have isomorphic subdivisions? Originally stated only for manifolds. No (Milnor, 1961) Examples of homeomorphic nonmanifold compact polyhedra which are not PL homeomorphic. Manifold Hauptvermutung Is every homeomorphism of compact PL manifolds homotopic to a PL homeomorphism? No: by the K.S. PLTOP smoothing theory.
9 9 TOP bundle theory TOP analogues of vector bundles and PL bundles. Microbundles = TOP bundles, with classifying spaces BTOP(n), BTOP = lim n BTOP(n). (Milnor, Kister 1964) A TOP manifold M n has a TOP tangent bundle τ M : M BTOP(n). For large k 1 M R k has a PL structure if and only if τ M : M BTOP lifts to a PL bundle τ M : M BPL.
10 10 DIFFPL smoothing theory DIFF structures on PL manifolds (Cairns, Whitehead, Hirsch, Milnor, Munkres, Lashof, Mazur,..., ) The DIFF structures on a compact PL manifold M are in bijective correspondence with the lifts of τ M : M BPL to a vector bundle τ M : M BO, i.e. with [M, PL/O]. Fibration sequence of classifying spaces PL/O BO BPL B(PL/O). The difference between DIFF and PL is quantified by { θ n for n 7 π n (PL/O) = 0 for n 6 with θ n the finite KervaireMilnor group of exotic spheres.
11 11 PLTOP smoothing theory PL structures on TOP manifolds (K.S., 1969) For n 5 the PL structures on a compact ndimensional TOP manifold M are in bijective correspondence with the lifts of τ M : M BTOP to τ M : M BPL, i.e. with [M, TOP/PL]. Fibration sequence of classifying spaces TOP/PL BPL BTOP B(TOP/PL) The difference between PL and TOP is quantified by { Z 2 for n = 3 π n (TOP/PL) = 0 for n 3 detected by the Rochlin signature invariant.
12 Signature The signature σ(m) Z of a compact oriented 4kdimensional ANR homology manifold M 4k with M = or a homology (4k 1)sphere Σ is the signature of the Poincaré duality nonsingular symmetric intersection form φ : H 2k (M) H 2k (M) Z ; (x, y) x y, [M] Theorem (Hirzebruch, 1953) For a compact oriented DIFF manifold M 4k σ(m) = L k (M), [M] Z with L k (M) H 4k (M; Q) a polynomial in the Pontrjagin classes p i (M) = p i (τ M ) H 4i (M). L 1 (M) = p 1 (M)/3. Signature theorem also in the PL category. Define p i (M), L i (M) H 4i (M; Q) for a PL manifold M n by L i (M), [N] = σ(n) Z for compact PL submanifolds N 4i M n R k with trivial normal PL bundle (Thom, 1958). 12
13 The signature mod 8 Theorem (Milnor, 1958 ) If M 4k is a compact oriented 4kdimensional ANR homology manifold with even intersection form then φ(x, x) 0 (mod 2) for x H 2k (M) σ(m) 0 (mod 8). 13 ( ) For a TOP manifold M 4k φ(x, x) = v 2k (ν M ), x [M] Z 2 for x H 2k (M) with v 2k (ν M ) H 2k (M; Z 2 ) the 2k th Wu class of the stable normal bundle ν M = τ M : M BTOP. So condition ( ) is satisfied if v 2k (ν M ) = 0. ( ) is satisfied if M is almost framed, meaning that ν M is trivial on M {pt.}. For k = 1 spin w 2 = 0 v 2 = 0 = ( ).
14 14 E 8 The E 8 form has signature E 8 = For k 2 let W 4k be the E 8 plumbing of 8 copies of τ S 2k, a compact (2k 1)connected 4kdimensional framed DIFF manifold with (H 2k (W ), φ) = (Z 8, E 8 ), σ(w ) = 8. The boundary W = Σ 4k 1 is an exotic sphere. The 4kdimensional nondiff almost framed PL manifold M 4k = W 4k Σ 4k 1 cσ obtained by coning Σ has σ(m) = 8.
15 Rochlin s Theorem Theorem (Rochlin, 1952) The signature of a compact 4dimensional spin PL manifold M has σ(m) 0(mod 16). The Kummer surface K 4 has σ(k) = 16. Every oriented 3dimensional PL homology sphere Σ is the boundary W of a 4dimensional framed PL manifold W. The Rochlin invariant α(σ) = σ(w ) 8Z/16Z = Z 2 accounts for the difference between PL and TOP manifolds! α(σ) = 1 for the Poincaré 3dimensional PL homology sphere Σ 3 = SO(3)/A 5 = W, with W 4 = the 4dimensional framed PL manifold with σ(w ) = 8 obtained by the E 8 plumbing of 8 copies of τ S 2. The 4dimensional homology manifold P 4 = W Σ cσ is homotopy equivalent to a compact 4dimensional spin TOP manifold M 4 = W Σ Q with Q 4 contractible, Q = Σ 3, (H 2 (M), φ) = (Z 8, E 8 ), σ(m) = 8 (Freedman, 1982). 15
16 The topological invariance of the rational Pontrjagin classes Theorem (Novikov, 1965) If h : M N is a homeomorphism of compact PL manifolds then h p i (N) = p i (M) H 4i (M; Q) (i 1). It suffices to prove the splitting theorem: for any k 1 and compact PL submanifold Y 4i N R k with π 1 (Y ) = {1} and trivial PL normal bundle the product homeomorphism h 1 : M R k N R k is proper homotopic to a PL map f : M R k N R k which is PL split at Y, meaning that it is PL transverse and f : X 4i = f 1 (Y ) Y is also a homotopy equivalence. Then L i (M), [X ] = σ(x ) = σ(y ) = L i (N), [Y ] Z, and h L i (N) = L i (M) H 4i (M; Q), so that h p i (N) = p i (M). 16
17 17 Splitting homotopy equivalences of manifolds For CAT = DIFF, PL or TOP define CAT isomorphism = diffeomorphism, PL homeomorphism, homeomorphism. A homotopy equivalence of CAT manifolds h : M N is CAT split along a CAT submanifold Y N if h is homotopic to a map f : M N CAT transverse at Y, with the restriction f : X = f 1 (Y ) Y also a homotopy equivalence of CAT manifolds. If h is homotopic to a CAT isomorphism then h is CAT split along every CAT submanifold. Converse: if h is not CAT split along one CAT submanifold then h is not homotopic to a CAT isomorphism!
18 The splitting theorem Theorem (Novikov 1965) Let k 1, n 5. If N n is a compact ndimensional PL manifold with π 1 (N) = {1}, W n+k is a noncompact PL manifold, h : W N R k is a homeomorphism, then h is PL split along N {0} N R k, with h proper homotopic to a PL transverse map f such that f : X = f 1 (N {0}) N is a homotopy equivalence. Proof: Wrap up the homeomorphism h : W N R k of noncompact simplyconnected PL manifolds to a homeomorphism g = h : V N T k of compact nonsimplyconnected PL manifolds such that h g : W = Ṽ N Rk. PL split g by kfold iteration of codim. 1 PL splittings along T 0 = {pt.} T 1 T 2 T k. Lift to PL splitting of h. The PL splitting needs the algebraic Ktheory computation K 0 (Z[Z k 1 ]) = 0, or BassHellerSwan Wh(Z k ) = 0. The kfold iteration of the Siebenmann (1965) end obstruction (unknown to N.) 18
19 The Stable Homeomorphism and Annulus Theorems 19 A homeomorphism h : M M is stable if h = h 1 h 2... h k : M M is the composite of homeomorphisms h i : M M each of which is the identity on an open subset U i R n. Stable Homeomorphism Theorem (Kirby, 1969) For n 5 every orientationpreserving homeomorphism h : R n R n is stable. Annulus Theorem (Kirby, 1969) If n 5 and h : D n D n is a homeomorphism such that h(d n ) D n S n 1 the homeomorphism 1 h : S n 1 S n 1 S n 1 h(s n 1 ) extends to a homeomorphism S n 1 [0, 1] = D n h(d n ).
20 Wrapping up and unwrapping Kirby s proof of the Stable Homeomorphism Theorem involves both wrapping up and unwrapping compact nonsimplyconnected T n wrapping up unwrapping noncompact simplyconnected R n The wrapping up passes from the homeomorphism h : R n R n to a homeomorphism h : T n T n using geometric topology, via an immersion T n {pt.} R n. Also need the vanishing of the end obstruction for π 1 = {1}. h is a bounded distance from 1 : T n T n, and hence stable. The unwrapping passes from h back to h using the surgery theory classification of PL manifolds homotopy equivalent to T n for n 5 via the algebraic Ltheory of Z[π 1 (T n )] = Z[Z n ]. 20
21 21 The original wrapping up/unwrapping diagrams From Kirby s 1969 Annals paper From the KirbySiebenmann 1969 AMS Bulletin paper
22 TOP/PL Theorem (K.S., 1969) Fibration sequence TOP/PL K(Z 2, 3) BPL κ BTOP B(TOP/PL) K(Z 2, 4) The Pontrjagin classes p k (η) H 4k (X ; Q) for TOP bundles η : X BTOP are defined by pullback from universal classes p k H 4k (BTOP; Q) = H 4k (BPL; Q). The Lgenus of a TOP manifold M n is defined by L k (M) = L k (τ M ) H 4k (M; Q), and for n = 4k σ(m) = L k (M), [M] Z. Bundles over S 4 classified by p 1 2Z H 4 (S 4 ; Q) = Q and κ H 4 (S 4 ; Z 2 ) = Z 2, with isomorphisms π 4 (BPL) = Z ; η p 1 ( η)/2, π 4 (BTOP) = Z Z 2 ; η (p 1 (η)/2, κ(η)). 22
23 TOP bundles over S 4 A TOP bundle η : S 4 BTOP has a PL lift η : S 4 BPL if and only if κ(η) = 0 H 4 (S 4 ; Z 2 ) = Z 2. A TOP bundle η : S 4 BTOP is fibre homotopy trivial if and only if J(η) = 0 π 4 (BG) = π S 3 = Z 24 or equivalently p 1 (η)/2 12κ(η) (mod 24). A fibre homotopy trivial TOP bundle η : S 4 BTOP has a PL lift η : S 4 BPL if and only if p 1 (η) 0(mod 48). The Poincaré homology sphere Σ 3 is used to construct a nonpl homeomorphism h : R n S 3 R n S 3 (n 4) with ph = p : R n S 3 S 3. The TOP(n)bundle η : S 4 BTOP(n) with clutching function h is fibre homotopy trivial but does not have a PL lift, with p 1 (η) = 24 Z, κ(η) = 1 Z 2. 23
24 PL structures on TOP manifolds 24 The PL structure obstruction of a compact ndimensional TOP manifold M κ(m) [M, B(TOP/PL)] = H 4 (M; Z 2 ) is the PL lifting obstruction of the stable tangent bundle τ M κ(m) : M τm BTOP κ B(TOP/PL) K(Z 2, 4). For n 5 κ(m) = 0 if and only if M has a PL structure. (K.S. 1969) If n 5 and κ(m) = 0 the PL structures on M are in bijective correspondence with [M, TOP/PL] = H 3 (M; Z 2 ). For each n 4 there exist compact ndimensional TOP manifolds M with κ(m) 0. Such M do not have a PL structure, and are counterexamples to the Combinatorial Triangulation Conjecture. All known counterexamples for n 5 can be triangulated.
25 The triangulation obstruction Rochlin invariant map α fits into short exact sequence 25 0 ker(α) θ H 3 α Z 2 0 with θ3 H homology spheres. the cobordism group of oriented 3dimensional PL ker(α) is infinitely generated (FintushelStern 1990, using Donaldson, 1982). (GalewskiStern, Matumoto, 1976) The triangulation obstruction of a compact ndimensional TOP manifold M is δκ(m) H 5 (M; ker(α)) with δ : H 4 (M; Z 2 ) H 5 (M; ker(α)) the Bockstein. For n 5 M can be triangulated if and only if δκ(m) = 0. Still unknown if δκ(m) can be nonzero for M n with n 5! M 4 with κ(m) 0 cannot be triangulated (Casson, 1985). E.g. the 4dim. Freedman E 8 manifold cannot be triangulated.
26 26 The handle straightening obstruction A homeomorphism h : M N of compact ndimensional PL manifolds has a handle straightening obstruction κ(h) = τ M h τ N [M, TOP/PL] = H 3 (M; Z 2 ). For n 5 κ(h) = 0 if and only if h is isotopic to a PL homeomorphism (K.S., 1969). The mapping cylinder of h is a TOP manifold W with a PL structure on boundary W = M N, such that W is homeomorphic to M [0, 1]. The handle straightening obstruction is the rel PL structure obstruction κ(h) = κ (W ) H 4 (W, W ; Z 2 ) = H 3 (M; Z 2 ). For each n 5 every element κ H 3 (M; Z 2 ) is κ = κ(h) for a homeomorphism h : M N.
27 TOP transversality 27 Theorem (K.S. 1970, RourkeSanderson 1970, Marin, 1977) Let (X, Y X ) be a pair of spaces such that Y has a TOP kbundle neighbourhood ν Y X : Y BTOP(k). For n k 4, every map f : M X from a compact ndimensional TOP manifold M is homotopic to a map g : M X which is TOP transverse at Y X, meaning that N n k = f 1 (Y ) M n is a codimension k TOP submanifold with normal TOP kbundle ν N M = f ν Y X : N BTOP(k) Also for n k = 4 (Quinn, 1988) TOP analogue of SardThom transversality for DIFF and PL, but much harder to prove.
28 28 TOP handlebodies Theorem (K.S. 1970) For n 6 every compact ndimensional TOP manifold M n has a handlebody decomposition M = h 0 h 1 h n with every ihandle h i = D i D n i attached to lower handles at + h i = S i 1 D n i h i. In particular, M is a finite CW complex. TOP analogue of handlebody decomposition for DIFF and PL, but much harder to prove. There is also a TOP analogue of Morse theory for DIFF and PL.
29 29 The TOP h and scobordism theorems An hcobordism is a cobordism (W ; M, N) such that the inclusions M W, N W are homotopy equivalences. TOP h and scobordism theorems (K.S. 1970). For n 5 an (n + 1)dimensional TOP hcobordism (W n+1 ; M, N) is homeomorphic to M ([0, 1]; {0}, {1}) rel M if and only if it is an scobordism, i.e. the Whitehead torsion is τ(m W ) = 0 Wh(π 1 (M)). If τ = 0 the composite homotopy equivalence M W N is homotopic to a homeomorphism. Generalization of the DIFF and PL cases originally due to Smale, 1962 and BardenMazurStallings, 1964.
30 Why are TOP manifolds harder than DIFF and PL manifolds? For CAT = DIFF and PL the structure theory of CAT manifolds can be developed working entirely in CAT to obtain transversality and handlebody decompositions. Need n 5 for Whitney trick for removing double points. But do not need sophisticated algebraic computation beyond Wh(1) = 0 required for the combinatorial invariance of Whitehead torsion. The highdimensional TOP manifold structure theory cannot be developed just in the TOP category! The TOP theory also needs the PL surgery classification of the homotopy types of the tori T n for n 5 which depends on the BassHellerSwan (1964) computation Wh(Z n ) = 0 or some controlled K or Ltheory analogue. 30
31 Why are TOP manifolds easier than DIFF and PL manifolds? Topological manifolds bear the simplest possible relation to their underlying homotopy types. This is a broad statement worth testing. L.C.Siebenmann (Nice ICM article, 1970) (R., 1992) The homotopy types of highdimensional TOP manifolds are in oneone correspondence with the homotopy types of Poincaré duality spaces with some additional chain level quadratic structure. Homeomorphisms correspond to homotopy equivalences preserving the additional structure. 31
32 Poincaré duality spaces 32 An ndimensional Poincaré duality space X is a space with the simple homotopy type of a finite CW complex, and a fundamental class [X ] H n (X ) such that cap product defines a simple chain equivalence [X ] : C(X ) n C(X ) inducing duality isomorphisms [X ] : H n (X ) = H (X ) with arbitrary Z[π 1 (X )]module coefficients. A compact ndimensional TOP manifold is an ndimensional Poincaré space (K.S., 1970). Any space homotopy equivalent to a Poincaré duality space is again a Poincaré duality space. There exist ndimensional Poincaré duality spaces which are not homotopy equivalent to compact ndimensional TOP manifolds (GitlerStasheff, 1965 and Wall, 1967 for PL, K.S for TOP)
33 33 The CAT manifold structure set Let CAT = DIFF, PL or TOP. The CAT structure set S CAT (X ) of an ndimensional Poincaré duality space X is the set of equivalence classes of pairs (M, f ) with M a compact ndimensional CAT manifold and f : M X a homotopy equivalence, with (M, f ) (M, f ) if there exists a CAT isomorphism h : M M with a homotopy f f h : M X. Fundamental problem of surgery theory: decide if S CAT (X ) is nonempty, and if so compute it by algebraic topology. This can be done for n 5, allowing the systematic construction and classification of CAT manifolds and homotopy equivalences using algebra.
34 The Spivak normal fibration 34 A spherical fibration η over a space X is a fibration S k 1 S(η) X e.g. the sphere bundle of a kplane vector or TOP bundle. Classifying spaces BG(k), BG = lim BG(k) with homotopy k groups the stable homotopy groups of spheres π n (BG) = π S n 1 = lim k π n+k 1 (S k ) The Spivak normal fibration ν X : X BG of an ndimensional Poincaré duality space X is S k 1 S(ν X ) = W W X (W, W ) regular neigbhd. of embedding X S n+k (k large). If M is a CAT manifold the Spivak normal fibration ν M : M BG lifts to the BCAT stable normal bundle ν CAT M : M BCAT of an embedding M S n+k (k large).
35 Surgery obstruction theory Wall (1969) defined the algebraic Lgroups L n (A) of a ring with involution A. Abelian GrothendieckWitt groups of quadratic forms on based f.g. free Amodules and their automorphisms. 4periodic: L n (A) = L n+4 (A). Let CAT = DIFF, PL or TOP. A CAT normal map f : M X from a compact ndimensional CAT manifold M to an ndimensional Poincaré duality space X has f [M] = [X ] H n (X ) and ν M f νx CAT : M BCAT for a CAT lift νx CAT : X BCAT of ν X : X BG. The surgery obstruction of a CAT normal map f σ (f ) L n (Z[π 1 (X )]) is such that for n 5 σ (f ) = 0 if and only if f is CAT normal bordant to a homotopy equivalence. Same obstruction groups in each CAT. Also a rel version, with homotopy equivalences on the boundaries. 35
36 The surgery theory construction of homotopy equivalences of manifolds from quadratic forms 36 Theorem (Wall, 1969, for CAT = DIFF, PL, after K.S. also for TOP). For an ndimensional CAT manifold M with n 5 every element x L n+1 (Z[π 1 (M)]) is realized as the rel surgery obstruction x = σ (f ) of a CAT normal map (f ; 1, h) : (W ; M, N) M ([0, 1]; {0}, {1}) with h : N M a homotopy equivalence. Build W by attaching middledimensional handles to M I { W n+1 M [0, 1] h i if n + 1 = 2i = M [0, 1] h i h i+1 if n + 1 = 2i + 1 using x to determine the intersections and selfintersections. Interesting quadratic forms x lead to interesting homotopy equivalences h : N M of CAT manifolds!
37 G/CAT classifies CAT bundles with fibre homotopy trivialization. If X is a Poincaré duality space with CAT lift of ν X then [X, G/CAT ] = the set of cobordism classes of CAT normal maps f : M X. Abelian group π n (G/CAT ) for X = S n. G/PL and G/TOP The classifying spaces BPL, BTOP, BG for PL, TOP bundles and spherical fibrations fit into a braid of fibrations TOP/PL G/PL BPL G/TOP BTOP BG 37
38 38 The surgery exact sequence Theorem (B.N.S.W. for CAT = DIFF, PL, K.S. for TOP) Let X be an ndimensional Poincaré duality space, n 5. X is homotopy equivalent to a compact ndimensional CAT manifold if and only if there exists a lift of ν X : X BG to ν X : X BCAT for which the corresponding CAT normal map f : M X with νm CAT = f ν X : M BCAT has surgery obstruction σ (f ) = 0 L n (Z[π 1 (X )]). If X is a CAT manifold the structure set S CAT (X ) fits into the CAT surgery exact sequence of pointed sets L n+1 (Z[π 1 (X )]) S CAT (X ) [X, G/CAT ] L n (Z[π 1 (X )]).
39 The Manifold Hauptvermutung from the surgery point of view The TOP and PL surgery exact sequences of a compact ndimensional PL manifold M (n 5) interlock in a braid of exact sequences of abelian groups L n+1 (Z[π 1 (M)]) S TOP (M) H 4 (M; Z 2 ) S PL (M) [M, G/TOP] H 3 (M; Z 2 ) [M, G/PL] L n (Z[π 1 (M)]) 39 A homeomorphism h : M N is homotopic to a PL homeomorphism if and only if κ(h) ker(h 3 (M; Z 2 ) S PL (M)). [κ(h)] [M, G/PL] is the Hauptvermutung obstruction of Casson and Sullivan (19667)  complete for π 1 (M) = {1}.
40 40 Why is the TOP surgery exact sequence better than the DIFF and PL sequences? Because it has an algebraic model (R., 1992)! For any space X can define the algebraic surgery exact sequence of cobordism groups of quadratic Poincaré complexes... L n+1 (Z[π 1 (X )]) S n+1 (X ) H n (X ; L(Z)) A L n (Z[π 1 (X )]) S n (X )... with L(Z) a 1connective spectrum of quadratic forms over Z, and A the assembly map from the local generalized L(Z)coefficient homology of X to the global Ltheory of Z[π 1 (X )]. π (L(Z)) = L (Z) and S ({pt.}) = 0.
41 41 Quadratic Poincaré complexes An ndimensional quadratic Poincaré complex C over a ring with involution A is an Amodule chain complex C with a chain equivalence ψ : C n = Hom A (C, A) n C. L n (A) is the cobordism group of ndimensional quadratic Poincaré complexes C of based f.g. free Amodules with Whitehead torsion τ(ψ) = 0 K 1 (A). H n (X ; L(Z)) is the cobordism group of sheaves C over X of ndimensional quadratic Poincaré complexes over Z, with Verdiertype duality. Assembly A(C) = q! p! C with p : X X the universal cover projection, q : X {pt.}. S n+1 (X ) is the cobordism group of sheaves C over X of ndimensional quadratic Poincaré complexes over Z with the assembly A(C) a contractible quadratic Poincaré complex over Z[π 1 (X )].
42 The total surgery obstruction of a Poincaré duality space An ndim. P. duality space X has a total surgery obstruction s(x ) = C S n (X ) such that for n 5 s(x ) = 0 if and only if X is homotopy equivalent to a compact ndimensional TOP manifold. The stalks C(x) (x X ) of C are quadratic Poincaré complexes over Z measuring the failure of X to be an ndimensional homology manifold, with exact sequences H r (C(x)) H n r ({x}) H r (X, X {x}).... s(x ) = 0 if and only if stalks are coherently nullcobordant. For n 5 the difference between the homotopy types of ndimensional TOP manifolds and Poincaré duality spaces is measured by the failure of the functor {spaces} {Z 4 graded abelian groups} ; X L (Z[π 1 (X )]) to be a generalized homology theory. 42
43 The total surgery obstruction of a homotopy equivalence of manifolds A homotopy equivalence f : N M of compact ndimensional TOP manifolds has a total surgery obstruction s(f ) = C S n+1 (M) such that for n 5 s(f ) = 0 if and only if f is homotopic to a homeomorphism. The stalks C(x) (x M) of C are quadratic Poincaré complexes over Z measuring the failure of f to have acyclic point inverses f 1 (x), with exact sequences H r (C(x)) H r (f 1 {x}) H r ({x}).... s(f ) = 0 if and only if the stalks are coherently nullcobordant. Theorem (R., 1992) The TOP surgery exact sequence is isomorphic to the algebraic surgery exact sequence. Bijection 43 S TOP (M) = S n+1 (M) ; (N, f ) s(f ).
44 44 Manifolds with boundary For an ndimensional CAT manifold with boundary (X, X ) let S CAT (X, X ) be the structure set of homotopy equivalences h : (M, M) (X, X ) with (M, M) a CAT manifold with boundary and h : M X a CAT isomorphism. For n 5 rel surgery exact sequence L n+k+1 (Z[π 1 (X )]) S CAT (X D k, (X D k )) [X D k, ; G/CAT, ] L n+k (Z[π 1 (X )]) L n+1 (Z[π 1 (X )]) S CAT (X, X ) [X, X ; G/CAT, ] L n (Z[π 1 (X )]). TOP case isomorphic to algebraic surgery exact sequence. Bijections S TOP (X D k, (X D k )) = S n+k+1 (X ) (k 0).
45 45 The algebraic Lgroups of Z (KervaireMilnor, 1963) The Lgroups of Z are given by Z (signature σ/8) L n (Z) = for n (mod 4) Z 2 (Arf invariant) Define the PL Lgroups of Z by { L n (Z) for n 4 Ln (Z) = {σ L 4 (Z) σ 0(mod16)} for n = 4 as in Rochlin s theorem, with L 4 (Z)/ L 4 (Z) = Z 2.
46 46 Spheres Generalized Poincaré Conjecture For n 4 a compact ndimensional TOP manifold M n homotopy equivalent to S n is homeomorphic to S n. For n 5: Smale (1960, DIFF ), Stallings (1961, PL), Newman (1962, TOP). For n = 4: Freedman (1982, TOP). For n + k 4 S TOP (S n D k, ) = S n+k+1 (S n ) = 0. π n (G/PL) = L n (Z) (Sullivan, 1967) π n (G/TOP) = L n (Z) (K.S., 1970), so L 0 (Z) G/TOP.
47 47 Simplyconnected surgery theory Theorem (K.S., 1970) For n 5 a simplyconnected ndimensional Poincaré duality space X is homotopy equivalent to a compact ndimensional TOP manifold if and only if the Spivak normal fibration ν X : X BG lifts to a TOP bundle ν X : X BTOP. TOP version of original DIFF theorem of Browder, Also true for n = 4 by Freedman, Corollary For n 5 a homotopy equivalence of simplyconnected compact ndimensional TOP manifolds h : M N is homotopic to a homeomorphism if and only if a canonical homotopy g : h ν N ν M : M BG lifts to a homotopy g : h ν TOP N ν TOP M : M BTOP.
48 48 For m, n 2, m + n 5 Products of spheres S PL (S m S n ) = L m (Z) L n (Z) S TOP (S m S n ) = S m+n+1 (S m S n ) = L m (Z) L n (Z) For CAT = PL and TOP there exist homotopy equivalences M m+n S m S n of CAT manifolds which are not CAT split, and so not homotopic to CAT isomorphisms. For CAT = PL these are counterexamples to the Manifold Hauptvermutung. There exist compact TOP manifolds M m+4 which are homotopy equivalent to S m S 4, but do not have a PL structure. Counterexamples to Combinatorial Triangulation Conjecture.
49 TOP/PL and homotopy structures 49 A map h : S k TOP(n)/PL(n) is represented by a homeomorphism h : R n D k R n D k such that ph = p : R n D k D k and which is a PL homeomorphism on R n S k 1. For n + k 6 can wrap up h to a homeomorphism h : M n+k T n D k with (M, M) PL, such that h : M T n S k 1 is a PL homeomorphism. Theorem (K.S., 1970) For 1 k < n, n 5 the wrapping up π k (TOP(n)/PL(n)) S PL (T n D k, ) ; h h is injective with image the subset S PL (T n D k, ) S PL (T n D k, ) invariant under transfers for finite covers T n T n, and π k (TOP(n)/PL(n)) = π k (TOP/PL). Key: approximate homeomorphism h : R n D k R n D k by homotopy equivalence h : M n+k T n D k.
50 The algebraic Lgroups of polynomial extensions 50 Theorem (Wall, Shaneson 1969 geometrically for A = Z[π], Novikov, R algebraically) For any ring with involution A L m (A[z, z 1 ]) = L m (A) L h m 1(A) with L h defined just like L but ignoring Whitehead torsion. Inductive computation n ( ) n L m (Z[Z n ]) = L m i (Z) i for any n 1, using Z[Z n ] i=0 = Z[Z n 1 ][z n, zn 1 ] = Z[z 1, z1 1, z 2, z2 1,..., z n, zn 1 ] and the BassHellerSwan computation Wh(Z n ) = 0.
51 Tori 51 Theorem (Wall, Hsiang, Shaneson 1969) [T n D k, ; G/PL, ] = n 1 ( ) n Ln+k i (Z) i=0 i L n+k (Z[Z n ]) = n ( ) n L n+k i (Z) (n, k 0), i=0 i ( ) S PL (T n D k, ) = H 3 k (T n n ; Z 2 ) = Z 2 (n + k 5) n + k 3 Corollary (K.S., 1970) For k < n and n 5 π k (TOP/PL) = S PL (T n D k, ) = { Z 2 if k = 3 0 if k 3 so that TOP/PL K(Z 2, 3). Need S PL (T n D k, ) = 0 (k 3) for handle straightening. S TOP (T n D k, ) = S n+k+1 (T n ) = 0 for n + k 5.
52 A counterexample to the Manifold Hauptvermutung from the surgery theory point of view The morphism L n+1 (Z[Z n ]) = [T n D 1, ; G/TOP, ] [T n, TOP/PL] = H 3 (T n ; Z 2 ) is onto, so for any x 0 H 3 (T n ; Z 2 ) there exists an element y L n+1 (Z[Z n ]) with [y] = x. For n 5 realize y = σ (f ) as the rel surgery obstruction of a PL normal map (f ; 1, g) : (W n+1 ; T n, τ n ) T n (I ; {0}, {1}) with g : τ n T n homotopic to a homeomorphism h, and s(g) = κ(h) = x 0 S PL (T n ) = H 3 (T n ; Z 2 ). The homotopy equivalence g is not PL split at T 3 T n with x, [T 3 ] = 1 Z 2, since g 1 (T 3 ) = T 3 #Σ 3 with Σ 3 = Poincaré homology sphere with Rochlin invariant α(σ 3 ) = 1. g is not homotopic to a PL homeomorphism. 52
53 A counterexample to the Combinatorial Triangulation Conjecture from the surgery theory point of view 53 For x 0 H 3 (T n ; Z 2 ), y L n+1 (Z[Z n ]), n 5 use the PL normal map (f ; 1, g) : (W n+1 ; T n, τ n ) T n (I ; {0}, {1}) with g : τ n T n homotopic to a homeomorphism h to define a compact (n + 1)dimensional TOP manifold M n+1 = W /{x h(x) x τ n } with a TOP normal map F : M T n+1 such that σ (F ) = (y, 0) L n+1 (Z[Z n+1 ]) = L n+1 (Z[Z n ]) L n (Z[Z n ]). The combinatorial triangulation obstruction of M is νm TOP κ(m) = δ(x) 0 im(δ : H 3 (T n ; Z 2 ) H 4 (M; Z 2 )). : M BTOP does not have a PL lift, so M does not have a PL structure, and is not homotopy equivalent to a compact (n + 1)dimensional PL manifold.
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