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Differential topology from the point of view of simple homotopy theory and further remarks

Differential topology from the point of view of simple homotopy theory and further remarks CORRECTIONS TO : DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY AND FURTHER REMARKS by BARRY MAZUR (1) w x. Corrections to : Differential Topology from the Point of View of Simple Homotopy Theory and further remarks. The purpose of this note is to atone for some of the sins committed in [I]. Namely: Chapters III and IV: Lemma 3.2 page 19 is false, and even if it were true it would be a bad idea to pass to such equivalence classes of cell decompositions. In fact, the notion of equi- valence given on page 18 is unnatural. In expiation, I gave (hopefully!) the right definition of equivalence in [2]. It is that definition (Definition i I, sec, 5) which meshes well with the proofs of [I]. As a consequence, one must also modify the definition of neighborhood of chapter IV. This is done in sec. 8 of [3], and the terminology is changed from ".neighborhood " to " solid ", which is really more appropriate. Here is a sketch of the new definitions: I. D-isotopy or Isotopy of differentiable cell decompositions : An object of the form X={(Xo, ..., X~); % i= ~, ..., v} and projection maps r~ : X~-+I (i=o, ..., ~) where X i ----- X i_ 1 U~D mi � D m- mi � I and the ?i are differentiable imbeddings such that ODm~� ~i> ~Xi_ t (1) Research was supported in part by NSF-GP x2I 7. 11 82 BARRY MAZUR is commutative. (Thus Xis a one-parameter family of cell decompositions). Restricting everything to any tcI we obtain a differentiable cell decomposition Xt={(=o*{t}, r:~-l{t}, ..., =~l{t}); qq[rc-l{t})}. Two cell decompositions X0, X 1 linked by an isotopy will be called isotopic. 2. Expansion Equivalence. -- Two cell decompositions M0, M 1 are expansion equi- valent if there are cell decompositions M~ = Mil.J irrelevant additions (i=o, I) such that M 0 is isomorphic with M~. 3. D-equivalence: The equivalence relation generated by a) isotopy; b) re-ordering equivalence; c) expansion equivalence. Any D-equivalence gives rise to a unique isotopy class of differentiable isomorphisms which will again be called D-equivalences. (They are free: i.e. they do not preserve the decomposition, of course.) 2. Structure weakening: Consider the categories: C o : differentiable manifolds (with boundary); differential imbeddings. C H : topological spaces; homotopy classes of continuous maps. (In CI~ set a*X=X). Then we may form cell decompositions in each of the categories. A differentiable cell decomposition has already been defined; a cell decomposition of C H is defined similarly where the maps are of C a and a is replaced by a*. Thus we obtain two categories of cell decompositions (maps are inclusions), and a structure-weakening functor: p : Do-~D H. Any notion in D o has its weak counterpart in D H. 3. Let XeD H. An n-dimensional solid over X (in Do) is a pair (M, ~), where MeDo, = : pM~X is a DH-equivalence. An isomorphism V : (M1, =1) ~ (M2, 7~2) between two M-solids is a D-equivalence ~" : MI-+M2 giving rise to the diagram p'y pM1 .... + pM2 866 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 8 3 (commutative up to Ci~-isotopy). (Broken-line arrows mean that they do not preserve decompositions.) Let N'~(X) denote the set of isomorphism classes of m-solids over X. With the understanding that we are always studying cell decompositions, up to this equivalence, we need never deal with " cell-filtrations ", that horror defined in [i]. (The reader should therefore skip Chapters III, IV and, rather, consult [2].) Chapter VII: In chapter VII, page 37 line 2, change the hypotheses to read m>2, n>2. Otherwise, for n= o, I, 2, the existence of homotopy-isolation data would not insure the existence of isolation data. Consequently, one needs the extra hypothesis n>2 throughout chapter VII. Chapter VIII: (i) The key geometric result of the theory is lemma 8.3. Since the hypotheses of chapter VII have been strengthened, and all our definitions have been changed, we must take up the proof of this lemma again. There are a few things to notice. Namely, after our new definitions, we do not have the contravariant map/k, and therefore the statement of the lemma must be changed; and with the new hypotheses for Chapter VII, we must exclude the case k= 5 which therefore remains unsolved. Finally, in this corrected proof, a gap in the old one (pointed out to us by C. Zeeman) will be filled. (To obtain a suitable notion of the essentiality of this lemma for our theory, one should notice that it, coupled with Prop. 5.4 of page 3 o, yields the nonstable neighborhood theorem for k> dim K + 5 immediately.) Let i : K ~ KU~A"U~A"+i be an elementary expansion, and let i(k / : Nk(K) -~ Nk(K ") be the " irrelevant addition " map defined by lemma 8. I, p. 43. Lemma (new 8.3). -- If (i) k>max{dim K-k I, M-+-4} , (2) k>5, then i k is a bijective isomorphism. Proof. -- a) If ikM 0 is equivalent to ikM1, by definition, M0 and Mx are expansion- equivalent to equivalent cell decompositions. Hence they themselves are equivalent, and i k is injective. b) Let us show that i k is surjective. [I assert on line I2, p. 45, that since Ns.A~k(K*), N = (MoU,D "~ � D k-m) U,D "+1 � D ~-m-1. 11" 84 BARRY MAZUR That needs proof. Therefore, to begin:] Lemma x.- Assume: (~) k>m + 2; (~) k> 5. Then, if Na,Uk(K*), N can be written (up to equivalence) as: N = (M0 U~D" � D k-') U+D m + ~ � D ~ .... ~. Assume Lemma I for the moment. Let us prove i(k ) surjective in three cases: I) The case m>2: Then the techniques of chapter VII apply, and the argument of p. 45, 46 yield Ng@,M, (p. 46 , line ~o). II) The cases re=o, ~: Trivial for dimensional reasons. III) The case m=2, k_>6: Then (after Lemma i) N = (M0 k U~ D 2 x D q) U, D 3 x D q- x where k-- 2 ---- q> 4. Let ~=7:1(M0) ~7:l(~Mo). The map : S I-+ 0M o is null-homotopic since it is null homotopic in M 0. Since dim 0M 0 >_5, ~ is an unknotted imbedding. Let M~=MoUq, D2xD q. Consider the natural maps ~g0vS2 < - (0M0--im ~) vS~ < (0M0--im r U~ (D2 x {x}) -c 0Mx where x~OD q, and h is a homotopy equivalence which is the identity on 0M0--im ~, and a map of degree -b I from D 2 x{x} to S 2. Then these maps are all isomorphisms for ~2, and we obtain the following commutative diagram: '~,~ (8M1) ~ Z [~] (~)'~2 (aM0) i~ 1 9 i, r:2(M1) g Z[~]| (Z[~] is the integral group ring of =, regarded as an abelian group). The vertical maps are the natural ones. Let S~2(SMx) denote the homotopy class of ~. Then we have ix(S ) : 1.IeZ[~] cZ[=]| Consequently: (*) S= 1.I| for xe=2(SM0). Let f: 0D3~8'I1 be a differentiable map representing the homotopy class S such that f(OD 3) intersects the pole {o}xODqcOM1 exactly at one point p, and transversally at p. This is possible after (*). Since dim 8Mx>5, fmay be approximated 568 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 85 by an imbedding g. If the approximation is sufficiently close, we may be sure that g has exactly one polar intersection also, which is transversal. Since g is homotopic to -~, Theorem 2 of w 2 below applies, yielding the following result: ~ is differentiably isotopic to an imbedding ~ which has exactly one transversal polar intersection. Thus the arguments of p. 45, 46 again apply. Proof of lemma i. -- We may take N properly ordered, N_--MoU~(DmxD ~-m) .UI(D,,xDk-m H tnm+lvr~k-,~-~ 11 r~m+lxDk-,~-I (etc.) where (etc.) refers to the remaining handles. The handles (Dm x D ~-")i may be removed from the vicinity of (Dm� k-") r k-m by differentiable isotopy. Therefore we may reorder the attaching " UcD"xD k-'~ " to come after all the (D"xDk-m)i, which we may now " lump " into the M0, and write: N_=-- MoU~DmxDk-mU (Uf Dm+l X D/~-m-IU (1) f,'" "Uf, nDm+lxD~'n-1) U+ Dm +1 xDk-m-1 (etc.) We may also regularize the f. by isotopy so that (*) f/(~D m+l X D~ -"-1) rl o m x ~Ok-m cD m x ~D k-'' (where ~D'. c gD" refers to the upper or lower hemisphere in bD 0. Since K* is an elementary expansion of K, we have that there is a continuous map, OD"+~ x I~N such that (i) f[OD"+l x o=-~ (ii) f(~Dm+l x ~) N Dmx Dk-m =Dm Xp for some pebD k-m (which we may take in OD~.-m). Lemma 2. -- After a differentiable isotopy (Pt) of N, we may find a continuous map f satisfying (i), (ii) above (for + = P~q~) such that (**) f(eD "+ 'x I) [7 o x D~ -m-1 ----0 (j = I, ..., m). For simplicity, denote Pi----{o}xD~-"-i (the jth ,, pole "). Call an element in f(~D m+l X I) n Pi a " polar intersection " Assume Lemma 2, for the moment. If (**) is true, then by an " expansion isotopy " centered at the pole Pi, we may obtain: (***) f(~Dm+l X I) ['l D~ + 1 x D~.~-~-~ =0. Since f]ODm+lX{O}=~, we may then arrange +(ODm+l � D ~- ~-1) rl D -'+1 x D~-m- 1 =13. Then we may reorder N, having the " ~ "-handle glued prior to the "f "-handles. Moreover (***) assures us that we have a map f : 0D "+1 X I ~ Mo U~,D m X Dk-mUcD "+1 x D k-"-I 569 86 BARRY MAZUR This means that we finally find ourselves in the situation I blithely took to be the " given " in my original proof. Proof of Lemma 2. -- By choosing f in general position we may suppose the inter- section f(OD" + 1� I) f'l Pj transversal. Therefore there are only a finite number of polar intersections. Let us " remove " them, one at a time Removal of a Polar intersection {x}: aH Pj v3 Choose nonsingular paths: From 0Pj to x along Pi ~1 ~ " " From x to f(0Dm+1� along f(0Dm-l� I) ~'2" " " From yz(I ) to yl(o) along ON "~'3 " " " with these properties: I. Y1 should contain no other polar intersections (possible since dim Pj>i by (~)). 2. "(~ should be disjoint from the image of the singular set Sc0D"~+I� under the mapf(possible since dim S<2 (m + i)--k< (m + i)--2 by (e) and therefore S cannot separate x from 0Din+iX{o}). 3- The path ~'3 should be extendable to a path defined for o<t<2 and should be so that the circuit y = ~,lkJ ~,~ (J-f3 is null-homotopic in N. (Possible since ~I(N, ON) =o by hypotheses (I), (2) of the main lemma). Let ecN be a nonsingular 2-disc whose boundary is y (possible by (~)). Orient everything in a neighborhood of ecN. Let P~ : o<t<2 be a differentiable isotopy of (N, ON) possessing these properties: (i) Po= I. (ii) Pt has support in some small neighborhood of "r (iii) P~(y3(o))=y3(t), o<t<2. (iv) The intersection of A=Plf(0D'~+I� and 0P iin ON at ya(I) is transversal. Now up to isotopy there are precisely two such Pt's (since OP i has positive dimension by (0~)), corresponding to plus and minus intersection indices between A and 0P~ at -h(I). 570 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 8 7 ~P ( m+l -.*-- ~f(aD x{Ol ) Pt 'r3 -~ ........ ._,/ Choose that Pt which yields an intersection index different from the intersection index of Pj and J'(OD'~+tXI) at x. After that isotopy Pt, we have introduced a new polar intersection y. It has an opposite index to x, and a yields " homotopy-isolation data " Now we may replacefby a map g which differs fromfonly in a small neighborhood of P, the segment between f-1(x) and f-l(y) in f-lTz such that a) f=g on ODm+ixaI (therefore g obeys (i), (ii)); b) g has precisely two fewer polar intersections than f (namely x and y). One constructs g by means of a, but first we make a few remarks about how meets f(3D'~+l� at P. Again by (a), if ~ is chosen in " general position with respect to f(OD'~+i� ", the normal directions along from P will be transverse to f(OD m + ix I). (Remark: int <r may intersect f(0D m + 1X I) at a finite number of unavoidable points.) Now we shall modify f in a neighborhood R of f-iF " guided by ~ ". Let ~' be an enlargement of % as visualized below. Let ~a'='~U-f'. Let "h � QcP be a tubular neighborhood of Yi in P, and choose some product coordination ~' � Q of a neighborhood of ~' so that the image off in a neighborhood of T' may be described as ~,'� As indicated below, modify f as follows: (i) g=f outside R. (ii) g(R) = ,(~ � QU ~' � ~Q. A i before 571 88 BARRY MAZUR Notice that this g may have more se/f-intersections than f because of the paren- thetical remark above. The important thing is that g satisfies a), b) above, and is continuous. (ii) Since we have included a new hypothesis (i.e. (2): k>5) in lemma 8.3, we must include a new hypothesis: k> 5 in the nonstable neighborhood theorem (p. 47). w 2. Low Dimensional Intersections: Let A, B be compact differentiable manifolds, and f0, fl :A-~B differentiable imbeddings. Consider the following weakening of the ordinary notion of differentiable isotopy: Definition I. -- Let LcA, KcB be finite subcomplexes. Then fo is congruent to fl mod(L-~K) if: Given any regular neighborhood of K, McB, there exists a regular neighborhood of L, NcA, and a differentiable isotopy q0t :A~B such that a) %=fo; b) q~l(N) CM ; c) qh]A--N =~ [A--N. This is a weakening of differentiable isotopy and taking L =0, K =0 one gets exactly differentiable isotopy. Definition 2. --fo is congruent to fl modq (written: f0-f~(mod q)) for q>o an integer, if there are complexes K q, L q--t (of dimension q, q-- I respectively) such that f0-f~ mod(Lq-t-->Kq) 9 (Is this an equivalence relation? We have introduccd this notion to obtain the following theorem :) Theorem I. -- Assume dim A ~ 2, dim B = 5, and that fo is homotopic to fl. Then f0 -fl(mod ,) Proof. -- In this range of dimensions, if f0 and fl are homotopic, then they are regularly homotopic. Let ft (o~t<I) be the regular homotopy, f: A� Then we may assume that there are exactly a finite number of points Pl, 9 9 P2n at whichffails to be a differentiable isotopy, and the immersions f possess only double points. Consider a pair of double points { (p~, to)(P2, to)} and for simplicity of notation assume this to be the only pair. (Our proof works as well in general.) Set Pi={pi}� to]CA� for j=i,2. Find f', a CX-approximation to f which is equal to f except in a small neighborhood of PILJP2, and such that: a) f'lPj is a differentiable imbedding possessing a nonsingular jacobian for j = I, 2. b) f'(Pl, to)=f'(P1)['lf'(P2)=f'(P2, to). To obtain f' a differentiable imbedding on Pj is easy. To insure that it have a nonsingular jacobian involves a slight calculation: If Gn,,~ is the Grassman manifold 572 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 8 9 of n-planes in m-space, then dim G2.5 = 6, dim Gt.4 = 3. Consequently a path of 2-planes in 5-space may be C * approximated by one such that no 2-plane of that path contains a given line. To obtain b) is also easy. Then f' will be, again, a regular homotopy with only one pair of singular points: (Pl, to), (P2, to). Let Si=D~(e)� [o, t0--8 ] (j= I, 2) be tubular neighborhoods of Pi, small enough so that f is a differentiable imbedding on S s (possible by implicit function theorem, since the jacobian off is nonsingular on Pi). Now modifyf' tof" which is Ct-close, has all the nice properties off' and the further property: f"(A� SIUS2) does not intersect the lines f"(Pj). (Possible since 3-[- I<5") By compactness there are differentiable tubular neighborhoods R t of f"(Pt) such that f"(A� does not intersect the Rj. Since f" is a differentiable imbedding on S i we may cut Rj, S t down to smaller tubular neighborhoods R~ CRj, S; cSj where S; ---- D~(r215 [o, tl] , tl= t0--W , which are adapted to one another in the following sense: f " (St, ' 0S;) C (Ri, 'OR;) j=I,2. To do this, a suitable version of the tubular neighborhoods lemma must be used. We may conclude: (*) f"(A� does not intersect R[[.JR' 2. Notice: I. f~' is a differentiable isotopy as t ranges in [tx, I] (q=t0+8'). For simplicity of notation, set D'= D~(r [J D~(r ; R'= R't [.J R ~ . Then: 2. f;' : (A--int D', 0D') -+ (B--int R', OR') is a differentiable isotopy for te [o, tl]. 3. f;': (D', OD') ~ (R', OR') is a regular homotopy, which is a differentiable on 0D' and a differentiable imbedding for t = t 1 . After (3) we may apply the isotopy extension theorem (relative version) to obtain a differentiable isotopy g, : (D', 0D') -~- (R', OR') te [o, t,] such that a) g,, =d," b) gt I OD'-=fI'[ OD' 673 BARRY MAZUR 9 ~ Now set h~ :A->B to be the (not yet differentiable) isotopy o<t<I given by: a) h[(a) =f[(a) if aCD or t>tl b) h;(a) =gt(a) if a~D', t~tl. This isotopy is not yet differentiable at 0D'� [o, tl], but it may be smoothed. Let h t be a differentiable isotopy which is C~ to h[ and Cl-close except in a small neighborhood about 0D'. Set K l=f''(P1) Uf"(P2)- Then R' could have been chosen small enough so as to be contained in a regular neighborhood of K ~. The final differentiable isotopy h t is CLclose to the original f except in some small neighborhood of D' (i.e. some small neighborhood of L~ and hl(D' ) cR'. Consequently h t may be approximated by a differentiable isotopy q0 which is a congruence (mod[L : K]) between f0 and fl. Thus theorem I is proved. It will be used in the following application: Let Y3cZ5 be a compact submanifold. Let f,g :XZ-+Z 5 be homotopic imbeddings of the compact differentiable 2-manifold X 2 in Z s. Theorem 2. -- Suppose g(X 2) meets y3 transversally at k points (o<k< + oo). Then there is a differentiable isotopy f : X2-+Z 5 such that fo =f and fl(X 2) meets y3 transversally at exactly k points. Proof. -- By theorem I, f-g (rood i). Thus f=gmod(L~ We may assume first that L ~ does not intersect g-l(g(XZ)flY3) since these are both zero- dimensional sets which may be moved about by differentiable isotopy. We may also assume that K 1 doesn't intersect y3, after a slight C~-perturbation of y3, say. Let M, N be regular neighborhoods K 1, L ~ respectively such that a) M fl Y~ = 0 b) Nflg-~(g(X) NY)=0. Applying theorem I we obtain a differentiable isotopyf such that (i) L=f; (ii) f~(N) cM; (iii) fll (X--N) =g[ (X--N). We obtain the following string of equalities: fl(X) NY =fl(X-- N) flY =g(~X2--N) NY =g(X) flY (the first because f~(N)r'lY=O, after (ii) and a); the second after (iii); the third because g(N) flY=O, after b)). Theorem 2 is therefore proved. w 3- Correction to : Definition of Equivalence of Combinatorial Imbeddlngs. Let me take this opportunity to warn the reader of an error in [3]- Namely: p. 1 I, condition (iii) in w 9 is impossible to obtain in general. Rather, one gets a union of intervals. The proof of the main theorem, however, can still be carried out. One 574 DII~'FERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 9 l should do it in a more direct way, however. The function spaces introduced in w 15 are unnecessary. The reader is referred to the recent I.H.E.S. seminar of C. Zeeman ibr a complete theory of combinatorial isotopy, which makes [3] unnecessary. REFERENCES [x] B. MAZUR, Differential topology from the point of view of simple homotopy theory, PubI. math., I.H.E.S., n ~ 15 (1963). [2] --, :$1orse theory in three categories, Symposium in Honor of Marston Morse, (1963) , Princeton. [3] --, Definition of equivalence of combinatorial imbeddings, Publ. math., I.H.E.S., n ~ 3 (I959). Refu le 15 J~vrier 1964. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Differential topology from the point of view of simple homotopy theory and further remarks

Mazur, Barry
Publications mathématiques de l'IHÉS , Volume 22 (1) – Aug 6, 2007
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Springer Journals
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Copyright © 1964 by Publications mathématiques de l’I.H.É.S
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Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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0073-8301
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10.1007/BF02684691
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CORRECTIONS TO : DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY AND FURTHER REMARKS by BARRY MAZUR (1) w x. Corrections to : Differential Topology from the Point of View of Simple Homotopy Theory and further remarks. The purpose of this note is to atone for some of the sins committed in [I]. Namely: Chapters III and IV: Lemma 3.2 page 19 is false, and even if it were true it would be a bad idea to pass to such equivalence classes of cell decompositions. In fact, the notion of equi- valence given on page 18 is unnatural. In expiation, I gave (hopefully!) the right definition of equivalence in [2]. It is that definition (Definition i I, sec, 5) which meshes well with the proofs of [I]. As a consequence, one must also modify the definition of neighborhood of chapter IV. This is done in sec. 8 of [3], and the terminology is changed from ".neighborhood " to " solid ", which is really more appropriate. Here is a sketch of the new definitions: I. D-isotopy or Isotopy of differentiable cell decompositions : An object of the form X={(Xo, ..., X~); % i= ~, ..., v} and projection maps r~ : X~-+I (i=o, ..., ~) where X i ----- X i_ 1 U~D mi � D m- mi � I and the ?i are differentiable imbeddings such that ODm~� ~i> ~Xi_ t (1) Research was supported in part by NSF-GP x2I 7. 11 82 BARRY MAZUR is commutative. (Thus Xis a one-parameter family of cell decompositions). Restricting everything to any tcI we obtain a differentiable cell decomposition Xt={(=o*{t}, r:~-l{t}, ..., =~l{t}); qq[rc-l{t})}. Two cell decompositions X0, X 1 linked by an isotopy will be called isotopic. 2. Expansion Equivalence. -- Two cell decompositions M0, M 1 are expansion equi- valent if there are cell decompositions M~ = Mil.J irrelevant additions (i=o, I) such that M 0 is isomorphic with M~. 3. D-equivalence: The equivalence relation generated by a) isotopy; b) re-ordering equivalence; c) expansion equivalence. Any D-equivalence gives rise to a unique isotopy class of differentiable isomorphisms which will again be called D-equivalences. (They are free: i.e. they do not preserve the decomposition, of course.) 2. Structure weakening: Consider the categories: C o : differentiable manifolds (with boundary); differential imbeddings. C H : topological spaces; homotopy classes of continuous maps. (In CI~ set a*X=X). Then we may form cell decompositions in each of the categories. A differentiable cell decomposition has already been defined; a cell decomposition of C H is defined similarly where the maps are of C a and a is replaced by a*. Thus we obtain two categories of cell decompositions (maps are inclusions), and a structure-weakening functor: p : Do-~D H. Any notion in D o has its weak counterpart in D H. 3. Let XeD H. An n-dimensional solid over X (in Do) is a pair (M, ~), where MeDo, = : pM~X is a DH-equivalence. An isomorphism V : (M1, =1) ~ (M2, 7~2) between two M-solids is a D-equivalence ~" : MI-+M2 giving rise to the diagram p'y pM1 .... + pM2 866 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 8 3 (commutative up to Ci~-isotopy). (Broken-line arrows mean that they do not preserve decompositions.) Let N'~(X) denote the set of isomorphism classes of m-solids over X. With the understanding that we are always studying cell decompositions, up to this equivalence, we need never deal with " cell-filtrations ", that horror defined in [i]. (The reader should therefore skip Chapters III, IV and, rather, consult [2].) Chapter VII: In chapter VII, page 37 line 2, change the hypotheses to read m>2, n>2. Otherwise, for n= o, I, 2, the existence of homotopy-isolation data would not insure the existence of isolation data. Consequently, one needs the extra hypothesis n>2 throughout chapter VII. Chapter VIII: (i) The key geometric result of the theory is lemma 8.3. Since the hypotheses of chapter VII have been strengthened, and all our definitions have been changed, we must take up the proof of this lemma again. There are a few things to notice. Namely, after our new definitions, we do not have the contravariant map/k, and therefore the statement of the lemma must be changed; and with the new hypotheses for Chapter VII, we must exclude the case k= 5 which therefore remains unsolved. Finally, in this corrected proof, a gap in the old one (pointed out to us by C. Zeeman) will be filled. (To obtain a suitable notion of the essentiality of this lemma for our theory, one should notice that it, coupled with Prop. 5.4 of page 3 o, yields the nonstable neighborhood theorem for k> dim K + 5 immediately.) Let i : K ~ KU~A"U~A"+i be an elementary expansion, and let i(k / : Nk(K) -~ Nk(K ") be the " irrelevant addition " map defined by lemma 8. I, p. 43. Lemma (new 8.3). -- If (i) k>max{dim K-k I, M-+-4} , (2) k>5, then i k is a bijective isomorphism. Proof. -- a) If ikM 0 is equivalent to ikM1, by definition, M0 and Mx are expansion- equivalent to equivalent cell decompositions. Hence they themselves are equivalent, and i k is injective. b) Let us show that i k is surjective. [I assert on line I2, p. 45, that since Ns.A~k(K*), N = (MoU,D "~ � D k-m) U,D "+1 � D ~-m-1. 11" 84 BARRY MAZUR That needs proof. Therefore, to begin:] Lemma x.- Assume: (~) k>m + 2; (~) k> 5. Then, if Na,Uk(K*), N can be written (up to equivalence) as: N = (M0 U~D" � D k-') U+D m + ~ � D ~ .... ~. Assume Lemma I for the moment. Let us prove i(k ) surjective in three cases: I) The case m>2: Then the techniques of chapter VII apply, and the argument of p. 45, 46 yield Ng@,M, (p. 46 , line ~o). II) The cases re=o, ~: Trivial for dimensional reasons. III) The case m=2, k_>6: Then (after Lemma i) N = (M0 k U~ D 2 x D q) U, D 3 x D q- x where k-- 2 ---- q> 4. Let ~=7:1(M0) ~7:l(~Mo). The map : S I-+ 0M o is null-homotopic since it is null homotopic in M 0. Since dim 0M 0 >_5, ~ is an unknotted imbedding. Let M~=MoUq, D2xD q. Consider the natural maps ~g0vS2 < - (0M0--im ~) vS~ < (0M0--im r U~ (D2 x {x}) -c 0Mx where x~OD q, and h is a homotopy equivalence which is the identity on 0M0--im ~, and a map of degree -b I from D 2 x{x} to S 2. Then these maps are all isomorphisms for ~2, and we obtain the following commutative diagram: '~,~ (8M1) ~ Z [~] (~)'~2 (aM0) i~ 1 9 i, r:2(M1) g Z[~]| (Z[~] is the integral group ring of =, regarded as an abelian group). The vertical maps are the natural ones. Let S~2(SMx) denote the homotopy class of ~. Then we have ix(S ) : 1.IeZ[~] cZ[=]| Consequently: (*) S= 1.I| for xe=2(SM0). Let f: 0D3~8'I1 be a differentiable map representing the homotopy class S such that f(OD 3) intersects the pole {o}xODqcOM1 exactly at one point p, and transversally at p. This is possible after (*). Since dim 8Mx>5, fmay be approximated 568 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 85 by an imbedding g. If the approximation is sufficiently close, we may be sure that g has exactly one polar intersection also, which is transversal. Since g is homotopic to -~, Theorem 2 of w 2 below applies, yielding the following result: ~ is differentiably isotopic to an imbedding ~ which has exactly one transversal polar intersection. Thus the arguments of p. 45, 46 again apply. Proof of lemma i. -- We may take N properly ordered, N_--MoU~(DmxD ~-m) .UI(D,,xDk-m H tnm+lvr~k-,~-~ 11 r~m+lxDk-,~-I (etc.) where (etc.) refers to the remaining handles. The handles (Dm x D ~-")i may be removed from the vicinity of (Dm� k-") r k-m by differentiable isotopy. Therefore we may reorder the attaching " UcD"xD k-'~ " to come after all the (D"xDk-m)i, which we may now " lump " into the M0, and write: N_=-- MoU~DmxDk-mU (Uf Dm+l X D/~-m-IU (1) f,'" "Uf, nDm+lxD~'n-1) U+ Dm +1 xDk-m-1 (etc.) We may also regularize the f. by isotopy so that (*) f/(~D m+l X D~ -"-1) rl o m x ~Ok-m cD m x ~D k-'' (where ~D'. c gD" refers to the upper or lower hemisphere in bD 0. Since K* is an elementary expansion of K, we have that there is a continuous map, OD"+~ x I~N such that (i) f[OD"+l x o=-~ (ii) f(~Dm+l x ~) N Dmx Dk-m =Dm Xp for some pebD k-m (which we may take in OD~.-m). Lemma 2. -- After a differentiable isotopy (Pt) of N, we may find a continuous map f satisfying (i), (ii) above (for + = P~q~) such that (**) f(eD "+ 'x I) [7 o x D~ -m-1 ----0 (j = I, ..., m). For simplicity, denote Pi----{o}xD~-"-i (the jth ,, pole "). Call an element in f(~D m+l X I) n Pi a " polar intersection " Assume Lemma 2, for the moment. If (**) is true, then by an " expansion isotopy " centered at the pole Pi, we may obtain: (***) f(~Dm+l X I) ['l D~ + 1 x D~.~-~-~ =0. Since f]ODm+lX{O}=~, we may then arrange +(ODm+l � D ~- ~-1) rl D -'+1 x D~-m- 1 =13. Then we may reorder N, having the " ~ "-handle glued prior to the "f "-handles. Moreover (***) assures us that we have a map f : 0D "+1 X I ~ Mo U~,D m X Dk-mUcD "+1 x D k-"-I 569 86 BARRY MAZUR This means that we finally find ourselves in the situation I blithely took to be the " given " in my original proof. Proof of Lemma 2. -- By choosing f in general position we may suppose the inter- section f(OD" + 1� I) f'l Pj transversal. Therefore there are only a finite number of polar intersections. Let us " remove " them, one at a time Removal of a Polar intersection {x}: aH Pj v3 Choose nonsingular paths: From 0Pj to x along Pi ~1 ~ " " From x to f(0Dm+1� along f(0Dm-l� I) ~'2" " " From yz(I ) to yl(o) along ON "~'3 " " " with these properties: I. Y1 should contain no other polar intersections (possible since dim Pj>i by (~)). 2. "(~ should be disjoint from the image of the singular set Sc0D"~+I� under the mapf(possible since dim S<2 (m + i)--k< (m + i)--2 by (e) and therefore S cannot separate x from 0Din+iX{o}). 3- The path ~'3 should be extendable to a path defined for o<t<2 and should be so that the circuit y = ~,lkJ ~,~ (J-f3 is null-homotopic in N. (Possible since ~I(N, ON) =o by hypotheses (I), (2) of the main lemma). Let ecN be a nonsingular 2-disc whose boundary is y (possible by (~)). Orient everything in a neighborhood of ecN. Let P~ : o<t<2 be a differentiable isotopy of (N, ON) possessing these properties: (i) Po= I. (ii) Pt has support in some small neighborhood of "r (iii) P~(y3(o))=y3(t), o<t<2. (iv) The intersection of A=Plf(0D'~+I� and 0P iin ON at ya(I) is transversal. Now up to isotopy there are precisely two such Pt's (since OP i has positive dimension by (0~)), corresponding to plus and minus intersection indices between A and 0P~ at -h(I). 570 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 8 7 ~P ( m+l -.*-- ~f(aD x{Ol ) Pt 'r3 -~ ........ ._,/ Choose that Pt which yields an intersection index different from the intersection index of Pj and J'(OD'~+tXI) at x. After that isotopy Pt, we have introduced a new polar intersection y. It has an opposite index to x, and a yields " homotopy-isolation data " Now we may replacefby a map g which differs fromfonly in a small neighborhood of P, the segment between f-1(x) and f-l(y) in f-lTz such that a) f=g on ODm+ixaI (therefore g obeys (i), (ii)); b) g has precisely two fewer polar intersections than f (namely x and y). One constructs g by means of a, but first we make a few remarks about how meets f(3D'~+l� at P. Again by (a), if ~ is chosen in " general position with respect to f(OD'~+i� ", the normal directions along from P will be transverse to f(OD m + ix I). (Remark: int <r may intersect f(0D m + 1X I) at a finite number of unavoidable points.) Now we shall modify f in a neighborhood R of f-iF " guided by ~ ". Let ~' be an enlargement of % as visualized below. Let ~a'='~U-f'. Let "h � QcP be a tubular neighborhood of Yi in P, and choose some product coordination ~' � Q of a neighborhood of ~' so that the image off in a neighborhood of T' may be described as ~,'� As indicated below, modify f as follows: (i) g=f outside R. (ii) g(R) = ,(~ � QU ~' � ~Q. A i before 571 88 BARRY MAZUR Notice that this g may have more se/f-intersections than f because of the paren- thetical remark above. The important thing is that g satisfies a), b) above, and is continuous. (ii) Since we have included a new hypothesis (i.e. (2): k>5) in lemma 8.3, we must include a new hypothesis: k> 5 in the nonstable neighborhood theorem (p. 47). w 2. Low Dimensional Intersections: Let A, B be compact differentiable manifolds, and f0, fl :A-~B differentiable imbeddings. Consider the following weakening of the ordinary notion of differentiable isotopy: Definition I. -- Let LcA, KcB be finite subcomplexes. Then fo is congruent to fl mod(L-~K) if: Given any regular neighborhood of K, McB, there exists a regular neighborhood of L, NcA, and a differentiable isotopy q0t :A~B such that a) %=fo; b) q~l(N) CM ; c) qh]A--N =~ [A--N. This is a weakening of differentiable isotopy and taking L =0, K =0 one gets exactly differentiable isotopy. Definition 2. --fo is congruent to fl modq (written: f0-f~(mod q)) for q>o an integer, if there are complexes K q, L q--t (of dimension q, q-- I respectively) such that f0-f~ mod(Lq-t-->Kq) 9 (Is this an equivalence relation? We have introduccd this notion to obtain the following theorem :) Theorem I. -- Assume dim A ~ 2, dim B = 5, and that fo is homotopic to fl. Then f0 -fl(mod ,) Proof. -- In this range of dimensions, if f0 and fl are homotopic, then they are regularly homotopic. Let ft (o~t<I) be the regular homotopy, f: A� Then we may assume that there are exactly a finite number of points Pl, 9 9 P2n at whichffails to be a differentiable isotopy, and the immersions f possess only double points. Consider a pair of double points { (p~, to)(P2, to)} and for simplicity of notation assume this to be the only pair. (Our proof works as well in general.) Set Pi={pi}� to]CA� for j=i,2. Find f', a CX-approximation to f which is equal to f except in a small neighborhood of PILJP2, and such that: a) f'lPj is a differentiable imbedding possessing a nonsingular jacobian for j = I, 2. b) f'(Pl, to)=f'(P1)['lf'(P2)=f'(P2, to). To obtain f' a differentiable imbedding on Pj is easy. To insure that it have a nonsingular jacobian involves a slight calculation: If Gn,,~ is the Grassman manifold 572 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 8 9 of n-planes in m-space, then dim G2.5 = 6, dim Gt.4 = 3. Consequently a path of 2-planes in 5-space may be C * approximated by one such that no 2-plane of that path contains a given line. To obtain b) is also easy. Then f' will be, again, a regular homotopy with only one pair of singular points: (Pl, to), (P2, to). Let Si=D~(e)� [o, t0--8 ] (j= I, 2) be tubular neighborhoods of Pi, small enough so that f is a differentiable imbedding on S s (possible by implicit function theorem, since the jacobian off is nonsingular on Pi). Now modifyf' tof" which is Ct-close, has all the nice properties off' and the further property: f"(A� SIUS2) does not intersect the lines f"(Pj). (Possible since 3-[- I<5") By compactness there are differentiable tubular neighborhoods R t of f"(Pt) such that f"(A� does not intersect the Rj. Since f" is a differentiable imbedding on S i we may cut Rj, S t down to smaller tubular neighborhoods R~ CRj, S; cSj where S; ---- D~(r215 [o, tl] , tl= t0--W , which are adapted to one another in the following sense: f " (St, ' 0S;) C (Ri, 'OR;) j=I,2. To do this, a suitable version of the tubular neighborhoods lemma must be used. We may conclude: (*) f"(A� does not intersect R[[.JR' 2. Notice: I. f~' is a differentiable isotopy as t ranges in [tx, I] (q=t0+8'). For simplicity of notation, set D'= D~(r [J D~(r ; R'= R't [.J R ~ . Then: 2. f;' : (A--int D', 0D') -+ (B--int R', OR') is a differentiable isotopy for te [o, tl]. 3. f;': (D', OD') ~ (R', OR') is a regular homotopy, which is a differentiable on 0D' and a differentiable imbedding for t = t 1 . After (3) we may apply the isotopy extension theorem (relative version) to obtain a differentiable isotopy g, : (D', 0D') -~- (R', OR') te [o, t,] such that a) g,, =d," b) gt I OD'-=fI'[ OD' 673 BARRY MAZUR 9 ~ Now set h~ :A->B to be the (not yet differentiable) isotopy o<t<I given by: a) h[(a) =f[(a) if aCD or t>tl b) h;(a) =gt(a) if a~D', t~tl. This isotopy is not yet differentiable at 0D'� [o, tl], but it may be smoothed. Let h t be a differentiable isotopy which is C~ to h[ and Cl-close except in a small neighborhood about 0D'. Set K l=f''(P1) Uf"(P2)- Then R' could have been chosen small enough so as to be contained in a regular neighborhood of K ~. The final differentiable isotopy h t is CLclose to the original f except in some small neighborhood of D' (i.e. some small neighborhood of L~ and hl(D' ) cR'. Consequently h t may be approximated by a differentiable isotopy q0 which is a congruence (mod[L : K]) between f0 and fl. Thus theorem I is proved. It will be used in the following application: Let Y3cZ5 be a compact submanifold. Let f,g :XZ-+Z 5 be homotopic imbeddings of the compact differentiable 2-manifold X 2 in Z s. Theorem 2. -- Suppose g(X 2) meets y3 transversally at k points (o<k< + oo). Then there is a differentiable isotopy f : X2-+Z 5 such that fo =f and fl(X 2) meets y3 transversally at exactly k points. Proof. -- By theorem I, f-g (rood i). Thus f=gmod(L~ We may assume first that L ~ does not intersect g-l(g(XZ)flY3) since these are both zero- dimensional sets which may be moved about by differentiable isotopy. We may also assume that K 1 doesn't intersect y3, after a slight C~-perturbation of y3, say. Let M, N be regular neighborhoods K 1, L ~ respectively such that a) M fl Y~ = 0 b) Nflg-~(g(X) NY)=0. Applying theorem I we obtain a differentiable isotopyf such that (i) L=f; (ii) f~(N) cM; (iii) fll (X--N) =g[ (X--N). We obtain the following string of equalities: fl(X) NY =fl(X-- N) flY =g(~X2--N) NY =g(X) flY (the first because f~(N)r'lY=O, after (ii) and a); the second after (iii); the third because g(N) flY=O, after b)). Theorem 2 is therefore proved. w 3- Correction to : Definition of Equivalence of Combinatorial Imbeddlngs. Let me take this opportunity to warn the reader of an error in [3]- Namely: p. 1 I, condition (iii) in w 9 is impossible to obtain in general. Rather, one gets a union of intervals. The proof of the main theorem, however, can still be carried out. One 574 DII~'FERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 9 l should do it in a more direct way, however. The function spaces introduced in w 15 are unnecessary. The reader is referred to the recent I.H.E.S. seminar of C. Zeeman ibr a complete theory of combinatorial isotopy, which makes [3] unnecessary. REFERENCES [x] B. MAZUR, Differential topology from the point of view of simple homotopy theory, PubI. math., I.H.E.S., n ~ 15 (1963). [2] --, :$1orse theory in three categories, Symposium in Honor of Marston Morse, (1963) , Princeton. [3] --, Definition of equivalence of combinatorial imbeddings, Publ. math., I.H.E.S., n ~ 3 (I959). Refu le 15 J~vrier 1964.

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Published: Aug 6, 2007

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