The algebraic surgery exact sequence ANDREW RANICKI (Edinburgh) - PDF document

The algebraic surgery exact sequence ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ � aa r • The algeb raic surgery exact sequence is de�ned fo r any space X ) A · · · → H n ( X ; L • − → L n ( Z [ π ( X )]) 1 → S n ( X ) → H n − ( X ; L • ) → . . . 1 with A the L -theo ry assembly map. The functo r X �→ S ∗ ( X ) is homotop y inva riant. The 2-stage obstructions of the Bro wder- • Novik ov-Sullivan-W all surgery theo ry fo r the existence and uniqueness of top ological man- ifold structures in a homotop y t yp e a re re- placed b y single obstructions in the relative groups S ∗ ( X ) of the assembly map A . 1

Lo cal and global mo dules • The assembly map A : H ∗ ( X ; L • ) → L ∗ ( Z [ π ( X )]) 1 is induced b y a fo rgetful functo r A : { ( Z , X )-mo dules } → { Z [ π ( X )]-mo dules } 1 where the domain dep ends on the lo cal top ology of X and the ta rget dep ends only on the fundamental group π ( X ) of X , which 1 is global. ! • In terms of sheaf theo ry A = q ! p with : � the universal covering p rojection p X → X : � and q X → { pt. } . • The geometric mo del fo r the L -theo ry as- sembly A is the fo rgetful functo r { geometric P oinca r � e complexes } → { top ological manifolds . } In fact, in dimensions n ≥ 5 this functo r has the same �b re as A . 2

Lo cal and global quadratic P oinca r � e complexes • (Global) The L -group L n ( Z [ π ( X )]) is the 1 cob o rdism group of n -dimensional quadratic P oinca r � e complexes ( C, ψ ) over Z [ π ( X )]. 1 • (Lo cal) The generalized homology group H n ( X ; L • ) is the cob o rdism group of n - dimensional quadratic P oinca r � e complexes ( C, ψ ) over ( Z , X ). As in sheaf theo ry C has stalks , which a re Z -mo dule chain com- plexes C ( x ) ( x ∈ X ). • (Lo cal-Global) S n ( X ) is the cob o rdism group of ( n − 1)-dimensional quadratic P oinca r � e complexes ( C, ψ ) over ( Z , X ) such that the Z [ π ( X )]-mo dule chain complex assembly 1 A ( C ) is acyclic. 3

The total surgery obstruction The total surgery obstruction of an • n -dimensional geometric P oinca r � e complex X is the cob o rdism class s ( X ) = ( C, ψ ) ∈ S n ( X ) of a Z [ π ( X )]-acyclic ( n − 1)-dimensional 1 quadratic P oinca r � e complex ( C, ψ ) over ( Z , X ). The stalks C ( x ) ( x ∈ X ) measure the fail- ure of X to have lo cal P oinca r � e dualit y )) → H n − r · · · → H r ( C ( x ( { x } ) → H r ( X, X \{ x } ) +1 )) → H n − r → H r − ( C ( x ( { x } ) → . . . 1 X is an n -dimensional homology manifold if and only if H ∗ ( C ( x )) = 0. In pa rticula r, this is the case if X is a top ological manifold. • T otal Surgery Obstruction Theo rem s ( X ) = 0 ∈ S n ( X ) if (and fo r n ≥ 5 only if ) X is homotop y equivalent to an n -dimensional top ological manifold. 4

The p ro of of the T otal Surgery Obstruction Theo rem The p ro of is a translation into algeb ra of the t w o-stage Bro wder-Novik ov-Sullivan-W all ob- struction fo r the existence of a top ological man- ifold in the homotop y t yp e of a P oinca r � e com- plex X : • The image t ( X ) ∈ H n − ( X ; L • ) of s ( X ) is 1 such that t ( X ) = 0 if and only if the Spi- vak no rmal �b ration ν X : X → BG has a top ological reduction � ν X : X → BTOP . If t ( X ) = 0 then s ( X ) ∈ S n ( X ) is the im- • age of the surgery obstruction σ ∗ ( f, b ) ∈ ( Z [ π ( X )]) of the no rmal map ( f : M → L n 1 X, b : ν M → � ν X ) determined b y a choice of lift � ν X : X → BTOP . • s ( X ) = 0 if and only if there exists a reduc- tion � ν X : X → BTOP fo r which σ ∗ ( f, b ) = 0. 5

The structure inva riant • The structure inva riant of a homotop y equiv- alence h : N → M of n -dimensional top o- logical manifolds is the cob o rdism class s ( h ) = ( C, ψ ) ∈ S n ( M ) +1 of a globally acyclic n -dimensional quadratic P oinca r � e complex ( C, ψ ). The stalks C ( x ) ( x ∈ M ) measure the failure of h to have acyclic p oint inverses, with ( h − 1 ( C ( x )) = H ∗ ( x ) → { x } ) H ∗ ( h − 1 � = H ∗ ( x )) ( x ∈ M ) . +1 • h has acyclic p oint inverses if and only if ( C ( x )) = 0. In pa rticula r, this is the H ∗ case if h is a homeomo rphism. • Structure Inva riant Theo rem s ( h ) = 0 ∈ S n ( M ) if (and fo r n ≥ 5 only +1 if ) h is homotopic to a homeomo rphism. 6

The p ro of of the Structure Inva riant Theo rem (I) The p ro of is a translation into algeb ra of the t w o-stage Bro wder-Novik ov-Sullivan-W all ob- struction fo r the uniqueness of top ological man- ifold structures in a homotop y t yp e : the image t ( h ) ∈ H n ( M ; L • ) of s ( h ) is such • that t ( h ) = 0 if and only if the no rmal in- va riant can b e trvialized ( h − 1 ) ∗ ν N − ν M ≃ {∗} : M → L 0 ≃ G/TOP if and only if 1 ∪ h : M ∪ N → M ∪ M extends to a no rmal b o rdism ( f, b ) : ( W ; M, N ) → M × ([0 , 1]; { 0 } , { 1 } ) • if t ( h ) = 0 then s ( h ) ∈ S n ( M ) is the +1 image of the surgery obstruction ( f, b ) ∈ L n ( Z [ π ( M )]) . σ ∗ +1 1 7

The p ro of of the Structure Inva riant Theo rem (I I) ( h ) = 0 if and only if there exists a no rmal • s b o rdism ( f, b ) which is a simple homotop y equivalence. • Have to w o rk with simple L -groups here, to tak e advantage of the s -cob o rdism the- o rem. • Alternative p ro of. The mapping cylinder of h : N → M = M ∪ h N × [0 , 1] P de�nes an ( n + 1)-dimensional geometric P oinca r � e pair ( P, M ∪ N ) with manifold b ound- a ry , such that P is homotop y equivalent to M . The structure inva riant is the rel ∂ to- tal surgery obstruction ( h ) = s ∂ ( P ) ∈ S n ( P ) = S n ( M ) . s +1 +1 8

The simply-connected case • F o r π ( X ) = { 1 } the algeb raic surgery ex- 1 act sequence b reaks up 0 → S n ( X ) → H n − ( X ; L • ) → L n − ( Z ) → 0 1 1 • The total surgery obstruction s ( X ) ∈ S n ( X ) maps injectively to the TOP reducibilit y ob- struction t ( X ) ∈ H n − ( X ; L • ) of the Spi- 1 vak no rmal �b ration ν X . Thus fo r n ≥ 5 a simply-connected n -dimensional geometric P oinca r � e complex X is homotop y equiva- lent to an n -dimensional top ological man- ifold if and only if ν X : X → BG admits a TOP reduction � ν X : X → BTOP . • The structure inva riant s ( h ) ∈ S n ( M ) +1 maps injectively to the no rmal inva riant ( h ) ∈ H n ( M ; L • ) = [ M, G/TOP ]. Thus fo r t n ≥ 5 h is homotopic to a homeomo rphism if and only if t ( h ) ≃ {∗} : M → G/TOP . 9

The geometric surgery exact sequence set S TOP • The structure ( M ) of a top olog- ical manifold M is the set of equivalence classes of homotop y equivalences h : N → M from top ological manifolds N , with h ∼ h ′ if there exist a homeomo rphism g : N ′ → N : N ′ → M y hg ≃ h ′ and a homotop . • Theo rem (Quinn, R.) The geometric surgery exact sequence fo r n = dim ( M ) ≥ 5 )]) → S TOP · · · → L n ( Z [ π ( M ( M ) 1 +1 → [ M, G/TOP ] → L n ( Z [ π ( M )]) 1 is isomo rphic to the relevant p o rtion of the algeb raic surgery exact sequence · · · → L n ( Z [ π ( M )]) → S n ( M ) 1 +1 +1 ) A → H n ( M ; L • − → L n ( Z [ π ( M )]) 1 with S TOP ( M × D k , M × S k − 1 ) = S n ( M ). + k +1 ∂ Example S TOP ( S n ( S n ) = S n ) = 0. +1 10

The image of the assembly map • Theo rem F o r any �nitely p resented group π the image of the assembly map A : H n ( K ( π, 1); L • ) → L n ( Z [ π ]) is the subgroup of L n ( Z [ π ]) consisting of the surgery obstructions σ ∗ ( f, b ) of no rmal maps ( f, b ) : N → M of closed n -dimensional manifolds with π ( M ) = π . 1 • There a re many calculations of the image of A fo r �nite π , notably the Oozing Con- jecture p roved b y Hambleton-Milgram-T a ylo r- Williams. 11