ALGEBRAIC AND GEOMETRIC SURGERY
CONTENTS CHAPTER 1 DI?ERENTIABLE MANIFOLDS 1 DI?ERENTIABLE MAPS ON EUCLIDEAN SPACES 2 SINGULAR POINTS AND MORSE FUNCTIONS 3 DI?ERENTIABLE MANIFOLDS AND DI?ERENTIABLE MAPS 4 DI?ERENTIABLE MANIFOLDS WITH BOUNDARY CHAPTER 2 BUNDLES 1 FIBRE BUNDLES 2 VECTOR BUNDLES 3 ASSOCIATED BUNDLES 4 REDUCTION OF THE GROUP OF A ?BRE BUNDLE 5 CLASSIFYING SPACES FOR ?BRE BUNDLES 6 THOM SPACES AND TRANSVERSALITY CHAPTER 3 IMMERSIONS AND EMBEDDINGS 1 EMBEDDINGS AND IMMERSIONS IN RM 2 CONSTRUCTING IMMERSIONS F : NN??MM WITH M ?2N 3 THE SINGULAR SET OF AN IMMERSION 4 IMMERSIONS OF SN IN R2N 5 THE WHITNEY TRICK CHAPTER 4 SIGNATURE INVARIANTS AND CHARACTERISTIC CLASSES 1 SOME TWO DIMENSIONAL EXAMPLES – THE FERMAT SURFACES 2 THE MULTISIGNATURE FOR FINITE GROUP ACTIONS 3 THE WALL EMBEDDING THEOREM CHAPTER 5 THE HOMOLOGY OF FIBRE BUNDLES 1 FOUR HOMOLOGY THEORIES 2 THE HOMOLOGY AND COHOMOLOGY OF A COVERING 3 POINCAR´E DUALITY 4 THE HOMOLOGY OF A ?BRATION CHAPTER 6 COBORDISM AND HANDLE DECOMPOSITIONS 1 FOUNDATIONS OF COBORDISM 2 HANDLES 3 LAGRANGIANS AND EVEN FORMS CHAPTER 7 THE HOMOTOPY AND PL CLASSI?CATION OF GENERALIZED LENS SPACES 1 THE HOMOTOPY CLASSI?CATION OF GENERALIZED LENS SPACES 2 THE TORSION OF CHAIN COMPLEXES WITH TRIVIAL HOMOLOGY 3 THE INVARIANCE OF TORSION UNDER CELLULAR SUBDIVISION 4 THE REIDEMEISTER-WHITEHEAD CLASSI?CATION OF LENS SPACES CHAPTER 8 SURGERY IN LOW DIMENSIONS 1 TYPES OF MODI?CATIONS OF MANIFOLDS 2 FURTHER EXAMPLES OF THE E?ECT OF SURGERY 3 THE E?ECT OF SURGERY ON HOMOLOGY 4 NORMAL MAPS AND SURGERY BELOW THE MIDDLE DIMENSION 5 DEGREE ONE NORMAL MAPS CHAPTER 9 SURGERY FOR SIMPLY CONNECTED MANIFOLDS 1 THE CASE N ?0 MOD (4) 2 THE CASE N = 4K + 2 AND ?1(M) = 3 SURGERY ON DEGREE ONE MAPS FOR N ODD AND ?1(X) = 4 THE GEOMETRIC MOVES 5 PLUMBING AND THE BROWDER-NOVIKOV THEOREM 6 THE BROWDER-NOVIKOV THEOREM 7 THE ARF-INVARIANT FOR SURGERY IN DIMENSION 4K + 2 IS WELL DE?NED 8 THE CLASSI?CATION OF HOMOTOPY SPHERES CHAPTER 10 THE ALGEBRAIC ANALYSIS OF SURGERY GROUPS WHEN ?1(X) = 1 THE VANISHING OF THE ODD SURGERY GROUPS WHEN ?1(X) = 2 TORSION FORMS AND SIGNATURE 3 THE CONSTRUCTION OF AN EXACT SEQUENCE FOR THE SURGERY GROUPS CHAPTER 11 THE GLOBAL STRUCTURE OF SURGERY WHEN ?1(X) = 1 THE SPIVAK NORMAL ?BRATION IN THE SIMPLY CONNECTED CASE 2 BROWDER’S EXTENSION OF THE SPIVAK NORMAL BUNDLE TO ?1(X) ?= 3 THE BASIC PROPERTIES OF HOMOTOPY SPHERE BUNDLES 4 THE SPIVAK BUNDLE AND DEGREE ONE NORMAL MAPS CHAPTER 12 SURGERY WHEN ?1(X) ?= {1} 1 PRELIMINARIES ON MODULES OVER RINGS WITH INVOLUTIONS 2 THE EVEN SURGERY OBSTRUCTION GROUPS 3 THE EVEN DIMENSIONAL SURGERY OBSTRUCTION 4 THE ODD DIMENSIONAL SURGERY OBSTRUCTION GROUPS 5 THE ODD DIMENSIONAL SURGERY OBSTRUCTION 6 REALIZING THE SURGERY OBSTRUCTIONS 7 SURGERY EXACT SEQUENCE AND BORDISM WHEN ?1(X) ?= CHAPTER 13 THE INSTANT SURGERY OBSTRUCTION AND PRODUCT FORMULAE 1 REVIEW OF ITERATED LOOP SPACE THEORY 2 THE QUADRATIC REDUCTION OF THE SURGERY KERNEL CHAPTER ?FM ?XJ HJ ¶ IF G: RM ?RS IS A SECOND DI?ERENTIABLE MAP AND GF : RN ?RS IS THE COMPOSITION THEN D(GF) = DG DF DEFINITION 1 LET F : RN ?RM BE A DI?ERENTIABLE MAP (1) A REGULAR POINT OF F IS A POINT X ?RN WHERE THE LINEAR MAP DF(X) IS OF MAXIMAL RANK, I E RANK(DF(X)) = MIN(M, N) (2) A CRITICAL POINT OF F IS A POINT X ?RN WHICH IS NOT REGULAR (3) A REGULAR VALUE OF F IS A POINT IN THE IMAGE Y ?RM SUCH THAT EVERY X ?F?1(Y) ?N IS REGULAR OR F?1(Y) IS EMPTY (4) A CRITICAL VALUE OF F IS A POINT Y ?RN WHICH IS NOT REGULAR THEOREM 1 (IMPLICIT FUNCTION) FOR ANY DI?ERENTIABLE MAP F : RN ? RM AND ANY REGULAR POINT X ?RN OF F THERE EXIST A NEIGHBORHOOD NX ?RN AND LOCAL COORDINATES (Z1, , ZN) IN NX WITH X AT (0, , 0), AND A NEIGHBOR- HOOD NF(X) ?RM OF F(X) WITH LOCAL COORDINATES (W1, , WM) WITH F(X) ? (0, , 0), SUCH THAT : IF N ?M, THEN F(Z1, , ZN) = (Z1, , ZN, 0, , | {Z } M?N TIMES ) IF N ?M THEN F(Z1, , ZN) = (Z1, , ZM) ?F ?XN ¶ ?RN , ALGEBRAIC AND GEOMETRIC SURGERY ¶ : RN × RN ?R ; (U, V) = ((U1, U2, , UN), (V1, V2, , VN)) ? P 1?I,J?N ?2F ?XI?XJ UIVJ , NAMELY µ : RN ?R ; H = (H1, H2, , HN) ??(H, H)/2 = P 1?I,J?N ?2F ?XI?XJ HIHJ WE SAY THAT THE CRITICAL POINT X OF THE FUNCTION F : RN ?R IS NON- DEGENERATE IF THE DETERMINANT OF THE HESSIAN MATRIX HF(X) = µ ?2F ?XI?XJ ¶ X IS NON-ZERO, DET(HF(X)) ?= 0 NOTE THAT IF WE CHANGE COORDINATES NEAR X SO (X1, , XN) = K(Y1, , YN) = (K1(Y1, , YN), , KN(Y1, , YN)) WITH DK INVERTIBLE AT X, THEN ?2(FH) ?YI?YJ = X R,S ?2F ?XR?XS ?KR ?YI ?KS ?YJ SINCE ?F ?XI = 0 AT X (1 ?I ?N) CONSEQUENTLY, HFK = (DK)THFDK AND HF ACTUALLY TRANSFORMS LIKE A SYMMETRIC BILINEAR FORM IN PARTICULAR : (1) NON-DEGENERACY FOR THE CRITICAL POINT X IS INVARIANT UNDER LOCAL CO- ORDINATE CHANGES, (2) SINCE WE CAN DIAGONALIZE A SYMMETRIC MATRIX BY AN ORTHOGONAL MA- TRIX, T, THERE IS AN ORTHOGONAL TRANSFORMATION CENTERED AT X WHICH ALGEBRAIC AND GEOMETRIC SURGERY N (FOR A PROOF SEE E G , MILNOR [43] ) REMARK IN PARTICULAR 1 HAS THE FOLLOWING IMPLICATIONS (1) NON-DEGENERATE CRITICAL POINTS ARE ISOLATED (2) A NON-DEGENERATE CRITICAL POINT OF INDEX N IS A LOCAL MAXIMUM (3) A NON-DEGENERATE CRITICAL POINT OF INDEX 0 IS A LOCAL MINIMUM NEAR A NON-DEGENERATE CRITICAL POINT WHERE THE INDEX I(F)X = K LIES PROPERLY BETWEEN 0 AND N THE GRAPH LOOKS LIKE A SADDLE POINT IT FOLLOWS THAT NEAR X THE LEVEL SURFACES, F?1(F(X) + ?) ?N(X) HAVE THE FORM ( SN?K?1 × DK FOR ? > FOR ? < 0 NEAR AN ISOLATED CRITICAL POINT THE HOMOTOPY TYPE OF THE LEVEL SPACE F?1(T) UNDERGOES A CHANGE IN HOMOTOPY TYPE NEAR X, F?1(F(X) + ?) HAS THE PROP- ERTY THAT IT LOOKS LOCALLY LIKE RN?1 FROM THE IMPLICIT FUNCTION THEOREM FOR ? ?= 0 BUT IT NEED NOT BE A COPY OF RN?1 IN ANY CASE, THE TYPE OF F?1(T) CHANGES NEAR X BY REMOVING A COPY OF DK × SN?K?1 (WHICH HAS BOUNDARY SK?1 × SN?K?1) FROM F?1(F(X) ??) AND REPLACING IT BY SK?1 × DN?K (WHICH ALGEBRAIC AND GEOMETRIC SURGERY HAS THE SAME BOUNDARY) F?1(F(X) ??) F?1(F(X) + ?) THE E?ECT ON THE LEVEL SURFACE OF PASSING THROUGH A CRITICAL POINT TO CLARIFY THE MEANING OF 2 ASSUMING THAT F?1(F(X) ??) IS SIMPLY A LINE SEGMENT NEAR X THEN THE NEW LEVEL SURFACE IS GIVEN AS THE LEVEL SURFACE AT F?1(F(X) + ?) DEFINITION 2 A C?FUNCTION F : RN ?R IS MORSE IF IT’S ONLY SINGU- LARITIES ARE NON-DEGENERATE CRITICAL POINTS MORSE PROVED THE FOLLOWING RESULT WHICH SHOWS THAT THE MORSE FUNC- TIONS ARE GENERIC AMONG ALL FUNCTIONS LEMMA 2 (MORSE DENSITY THEOREM) THE SET OF MORSE FUNCTIONS IS OPEN AND DENSE IN THE SET OF ALL C?FUNCTIONS F : RN ?R IN PARTICULAR, GIVEN ANY FUNCTION F : RN ?R, IT CAN BE UNIFORMLY AP- PROXIMATED BY A MORSE FUNCTION THE STRUCTURE OF F?1((??, T]) FOR F MORSE SUPPOSE THAT THE NUM- BER OF SINGULAR POINTS OF THE MORSE FUNCTION F IS ?NITE, OR MORE GENERALLY RESTRICT ATTENTION TO A COMPACT REGION NL ?RN, WHERE NL = {X ?RN | |X| ?L} ALGEBRAIC AND GEOMETRIC SURGERY THE FACT THAT THE SINGULAR POINTS ARE ISOLATED ASSURES THAT THERE ARE ONLY A ?NITE NUMBER OF CRITICAL POINTS HENCE, THERE ARE ONLY A ?NITE NUMBER IF T ?R, SAY T1 < T2 < < TK, WHICH ARE CRITICAL VALUES FOR F WE CONSIDER THE SPACES HF(T) = F?1((??, T]) AND ASK HOW THEY CHANGE AS T PASSES THROUGH THE CRITICAL VALUES OF F IN FACT, THIS IS THE ONLY TIME THE HF(T) CAN CHANGE SINCE WE HAVE LEMMA 3 SUPPOSE THAT THERE IS NO CRITICAL VALUE OF F IN THE INTERVAL [T1, T2] THEN HF(T1) IS DI?ERENTIABLY HOMEOMORPHIC TO HF(T2) PROOF THE IDEA HERE IS THAT THE GRADIENT VECTOR ?ELD DF DE?NES A ?OW ON RN, H(X, T) FOR WHICH F(H(X, T)) IS A MONOTONE INCREASING FUNCTION OF T AND IS STATIONARY ONLY AT THE CRITICAL POINTS OF F SINCE THERE ARE ?1, ?2 > 0 SO THAT THERE ARE NO CRITICAL VALUES OF F IN THE INTERVAL (T1 ??1, T2 + ?2) WE SQUEEZE THE GRADIENT ?ELD TO ZERO FOR F(X) ?T1 ??1 AND THEN USE THE RESULTING ?OW TO EXPAND HF(T1) TO IDENTIFY IT WITH HF(T2) ? LEMMA 4 SUPPOSE THAT T IS A CRITICAL VALUE OF F AND X1, , XR ARE THE CRITICAL POINTS WITH F(XI) = T SUPPOSE THAT THE INDEX OF F AT XI IS II THEN HF(T + ?) HAS THE HOMOTOPY TYPE OF HF(T ??) WITH ONE CELL EIJ ATTACHED FOR EACH CRITICAL POINT XJ OF INDEX IJ IN F?1(T) PROOF WE HAVE THAT HF(T + ?) IS IDENTI?ED WITH HF(T ??) OUTSIDE OF SU?CIENTLY SMALL NEIGHBORHOODS OF THE CRITICAL POINTS FROM THE PROOF OF 3 BUT IN A SMALL NEIGHBORHOOD OF XI WE HAVE THAT HF(T + ?) IS IDENTI?ED WITH HF(T ??) WITH THE DISK DK = ( 0, , (XI)2 ?? ATTACHED F?1(T ??) F?1(T ??) F?1(T + ?) F?1(T + ?) AS IT IS FAIRLY DIRECT FROM THE DIAGRAM ABOVE TO SEE THAT THE LEFT AND RIGHT ARCS ABOVE CORRESPOND THE THE INTERSECTION OF THE LEVEL SURFACE HF(T??) WITH THE NEIGHBORHOOD OF THE CRITICAL POINT XI, AND THE ENTIRE REGION DEFORMATION RETRACTS TO THE DISK ABOVE UNION WITH THE ORIGINAL LEVEL SET ? ALGEBRAIC AND GEOMETRIC SURGERY K = ) WE CAN COVER SM BY TWO OPEN SUBSETS U+ = SM\(1, 0, , 0) , U?= SM\(?1, 0, , 0) DE?NE A HOMEOMORPHISM ?+ : RM ?U+ BY ?+(X1, , XM) = (F(|X|), P 1 ?F(|X|)2 X |X|) WHERE F(T) IS A MONOTONE INCREASING C?FUNCTION OF T WITH F(0) = ?1 AND LIMT7??(F(T)) = 1 THERE IS A SIMILAR HOMEOMORPHISM ??: RM ?U?, BUT USING THE FUNCTION G(T) = ?F(T) IN THIS CASE THE INTERSECTION OF THE TWO NEIGHBORHOODS IS U+ ?U?= ?+(RM\{0}) AND THE TRANSITION FUNCTION ?+,?= (??)?1?+| : (?+)?1(U+ ?U?) = RM\{0} ?(??)?1(U+ ?U?) = RM\{0} IS GIVEN BY ?+,?(X) = P T/¯TX WHERE T = |X| AND F(¯T) = ?F(T) ALGEBRAIC AND GEOMETRIC SURGERY WITH (M + 1) CRITICAL POINTS [0, , 0, 1, 0, , 0] OF INDEX 0, 1, , M ALGEBRAIC AND GEOMETRIC SURGERY I (?M ??IRM + ??JRM +) IS CONTAINED IN THE POINTS OF RM + WITH XM = 0 THE PAIR (MM, ?M) IS A DI?ERENTIABLE MANIFOLD WITH ALGEBRAIC AND GEOMETRIC SURGERY