Hirsch, M.W., Differential Topology, Springer-Verlag, New York, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are subspaces of are ambient isotopic if there is a continuous mapping 2 # such that for each ( # , 0 is a homeomorphism from onto 3 4 5 1 is the identity and . 1768 . For other fundamental terms, the reader is referred to the text [22]. Although any two simple closed planar curves are ambient isotopic, Figure 1 shows two simple homeomorphic curves, which are not ambient isotopic , where the PL curve is an approximation of the curve on the left. In the right half of Figure 1 the : co ordinates of some vertices are specifically ....

....of differential topology, PL topology and knot theory are the most relevant, for which key summary references are given, below. In differential topology, extension of isotopies to ambient isotopies is accomplished on a manifold without boundary by constructing a tubular neighborhood [22]. The assumption of is natural within that context, but is not invoked here. Rather, our results only require continuity. From PL topology, there are necessary and sufficient conditions for an isotopy of compact polyhedra [32, Theorem 4.24, p. 58] to be extended to an ambient isotopy. A common ....

Hirsch, M.W., Differential Topology, Springer-Verlag, New York, 1976.

....concept than the notion of a connected set of fixed points with nonzero index. For manifolds the answer is no, as we show in Theorem 6. A version of this result is proved by O Neill (1953) and it appears that this 4 For the definition of a manifold (manifold with boundary) see Milnor (1965) or Hirsch (1976). In this paper a manifold may have an empty boundary, and the term manifold is reserved for boundaryless manifolds. 12 result explains the neglect of essential sets in the subsequent mathematical literature. Finally, we discuss in detail how the theory of essential sets can be used to prove ....

....a sequence (x n ; y n ) 2 (X Theta Y ) Gamma W with d(f(x n ) y n ) 0: Taking a subsequence, let x n x: Then (x; f(x) 2 Gr(f) is a limit point of (X Theta Y ) Gamma W; a contradiction. In general the relative topology of C(X; Y ) ae K(X; Y ) is the strong C 0 topology (Hirsch (1976, p. 35) but we will not need this fact. If X ae Y then the fixed point set of a correspondence F : X Y is F(F ) fx 2 X j x 2 F (x)g: If F 2 K(X; Y ) then Gr(F ) is closed in X Theta Y (Hildenbrand (1974, p. 24) so F(F ) is closed in X: If X and Y are compact and F n F in K(X; Y ) ....

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Hirsch, M.W. (1976): Differential Topology, Springer-Verlag, New York.

....small then F 1;ff is a C 1 perturbation of F , the autonomous dynamical system for ff = 0. If F is a diffeomorphism in a neighborhood of the attractor A 0 then F 1;ff is a diffeomorphism in a neighborhood of A ff because of the openness of the set of C 1 diffeomorphisms, e.g. see M.W. Hirsch [22], page 38. Hence, G 1;ff and G 2;ff are conjugate. F 1;ff (A ff ) is an attractor for G 2;ff homeomorphic to A ff and A ff [F 1;ff (A ff ) is an attractor for (3.1) the nonautonomous system. Notice that this process does not prove that A ff is homeomorphic to A 0 because there is no conjugacy for ....

M.W. Hirsch, Differential Topology, Springer, New York, 1976.

....differential topology, piecewise linear topology and knot theory are the most relevant, for which key summary references will be given. In differential topology, extension of isotopies to ambient isotopies is accomplished on a C 1 manifold without boundary by constructing a tubular neighborhood [23]. The assumption of C 1 is natural within that context, but is not invoked here. Rather, our results only require C 2 continuity. From piecewise linear topology topology, there are necessary and sufficient conditions for an isotopy of compact polyhedra [32, Theorem 4.24, p. 58] to be extended ....

Hirsch, M.W., Differential Topology, Springer-Verlag, New York, 1976.

....p ) 2 C 3 (IR n 1 ; IR p 1 ) fi fi fi fi fi each point of Sigma gc belongs to one of the Types 1; 2; 3; 4; 5 ) In [12] it is also shown that F is a C 3 s open and dense subset of the space C 3 (IR n 1 ; IR p 1 ) endowed with the strong (or Whitney ) C 3 s topology. c.f. e.g. [9] or [13] for the definition of this topology) If the one parametric problem (P (t) belongs to F , then, the corresponding set Sigma gc has a suitable structure for the use of pathfollowing methods. Pathfollowing methods are the main tools for solving one parametric optimization problems. ....

Hirsch, M.W., Differential topology, Grad. Texts Math., Vol. 33, SpringerVerlag Berlin, 1976.

....x is a vector, as x will be used to denote a (single) generic player. 8 A fiber bundle is a space E with a base space B E and a projection p : E B where the set p Gamma1 (b) is called the fiber over b 2 B. Note that in our notation Sigma(x) is used to denote p Gamma1 (x) See, e.g. [18] for more details about fiber bundles. 9 We say that a subbundle S is compact if for 8x 2 X the set S(x) is compact. 7 where Gamma x is a (measurable) function with decreasing differences: for all oe; oe 0 2 Sigma(x) and oe oe 0 Gamma x (oe; Gamma Gamma x (oe 0 ; is ....

M. W. Hirsch. Differential Topology. Springer-Verlag, New York, 1976.

....Let F be a smooth algebraic function defined on an open semialgebraic set U ae R n i and determined by a polynomial with coefficients from R i . Then i 1 is not a critical value of F (i.e. grad y (F ) does not vanish at any point y 2 fF = i 1 g U (R i 1 ) Proof. Sard s theorem [Hi 76] and the transfer principle imply the finiteness of the set of all critical values of F in U (R i 1 ) moreover this set lies in R i . 3. Curved points For any i face P i denote by P i the i plane containing P i . First let us reduce Theorem 2 to the case of compact P . Let t be the minimal ....

.... ae ff k1 1 0; f k 0g P i = W P i = W P i ae P i : Since dim(ff k1 1 0; f k 0g P i ) i, each connected component of the set ff k1 1 0; f k 0g P i contains a connected component of the smooth hypersurface fg = g P i (in P i ) due to Morse theory (see [Hi 76] and in view of (a) Moreover, each connected component of the hypersurface fg = g P i either lies completely in the set ff k1 1 0; f k 0g P i or does not intersect this set. Finally, the inequality dim Gamma cl(K 0 i n P i ) K 0 i P i Delta i Gamma 2 ....

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M. Hirsch "Differential Topology," Springer-Verlag, 1976.

.... Delta g (j) I j 0g: 3) Introduce a polynomial g = Y 1lI j g (j) l ; and choose a real 0 satisfying the following requirements: a) is smaller than the absolute value of any nonzero critical value of the restriction of g on Pi for any i facet Pi of P (by Sard s theorem [Hi 76] there exist only a finite number of critical values) b) polynomial g Gamma does not vanish identically on any irreducible component of every intersection V (l) Pi; 1 l r (there exists at most finite number of possible values of such that g Gamma vanishes identically on V (l) ....

....(a) implies (involving the implicit function theorem) that Pi fg = g is a nonsingular hypersurface in Pi. From the property (b) it follows that dim(fg = g V (l) Pi) i Gamma 1 (4) for each r 1 1 l r. Observe that, due to (a) and according to elementary facts from Morse theory [Hi 76] every connected component of the set V (l) Pi fg (j) 1 0 Delta Delta Delta g (j) I j 0g (see (3) contains at least one 6 (necessarily compact) connected component of the hypersurface fg = g in Pi 1 (note that the signs of all polynomials g (j) 1 ; g (j) I ....

M. Hirsch "Differential Topology," Springer--Verlag, 1976.

....assert that P is invariant, and is the union of immersed submanifolds of W s (a) and of W u (b) Moreover, the intersection of P with any fundamental domain is generically a neat submanifold. Recall that when M is a manifold with boundary, a set A ae M is neat in M if A = A M ; 4) cf. [28] for the definition) In other words, the boundary of the submanifold is nicely placed in the boundary of the manifold. For any fixed fundamental domain S, the primary intersection does not have to be a neat submanifold of S. However, if the intersection of the stable and unstable manifolds in ....

M.W. Hirsch. Differential Topology. Springer-Verlag, 1976.

.... Delta g (j) I j 0g: 3) Introduce a polynomial g = Y 1lI j g (j) l ; and choose a real 0 satisfying the following requirements: a) is smaller than the absolute value of any nonzero critical value of the restriction of g on Pi for any i facet Pi of P (by Sard s theorem [Hi 76] there exist only a finite number of critical values) b) polynomial g Gamma does not vanish identically on any irreducible component of every intersection V (l) Pi; 1 l r (there exists at most finite number of possible values of such that g Gamma vanishes identically on V (l) ....

....implies (involving the implicit function theorem) that Pi fg = g is a nonsingular hypersurface in Pi. From the property (b) it follows that dim(fg = g V (l) Pi) i Gamma 1 (4) for each r 1 1 l r. Observe that, due to (a) and according to elementary facts from Morse theory [Hi 76] every connected component of the set V (l) Pi fg (j) 1 0 Delta Delta Delta g (j) I j 0g (see (3) containes at least one (necessarily compact) connected component of the hypersurface fg = g in Pi 4 (note that the signs of all polynomials g (j) 1 ; g (j) I j ....

M. Hirsch "Differential Topology," Springer-Verlag, 1976.

....1 L 2 equivariant vector bundle Xi 00 M red W;E;L 1 then extends to an S 1 L 2 equivariant vector bundle r Xi 00 U , a subbundle of r ( ffi V 2 j M red W;E;L 1 ) U . Since the map r is a C 1 retraction, PU(2) MONOPOLES AND LINKS 43 there is a C 1 isomorphism [30], 32] f : ffi V 2 j U r ( ffi V 2 j M red W;E;L 1 ) Hence, we obtain a (trivial) C 1 bundle Xi 0 U a real subbundle Xi 0 j U ae ffi V 2 j U by setting Xi 0 : f Gamma1 (r Xi 00 ) By construction, the fiberwise L 2 orthogonal projection ffi V 2 ....

M. W. Hirsch, Differential topology, Springer, New York, 1976.

....e i =2 allows the branch cylinder to roll completely around once as varies from Gammato . 1 A C 2 diffeomorphism is a bijective mapping F such that both F and F Gamma1 are twice continuously differentiable. 2 and without loss of generality, since the Whitney embedding theorem ([8], ch.1, Theorem 3.5) implies that any manifold may be embedded in a Euclidean space BENEATH THE NOISE, CHAOS 3 Figure 1 For each ff the diffeomorphism F ff has an attractor ae Omega whose intersection with any slice Omega fi = f(e i ; z) fig is a Cantor set see Figure 2. For ....

M. Hirsch (1976) Differential Topology. Springer-Verlag.

.... i gc o : Definition 9 A one parametric variational inequality (F; H;G) 2 D belonging to V will be called regular (in the sense of Jongen, Jonker and Twilt) An important property of the regularity is its openness and its density with respect to the strong Whitney topology (see e.g. 1] [7], 9] Let us state the local openness in the following. Theorem 2 Let ( F ; H; G) 2 D and z 2 Sigma i gc ( F ; H; G) or z = 2 Sigma gc ( F ; H; G) i 2 f1; 5g. 23 Then there exist an open neighbourhood U z from z and a positive number r z such that Sigma gc ( ....

Hirsch M.W.: Differential Topology, Grad. Texts Math., Vol. 33, Springer Verlag, Berlin, 1976. 27

...., Gamma1 1. Let a(x; y) be any nonnegative smooth function which is zero exactly on the closure of the spiral S (that is, S plus the origin) Such a function always exists since any closed subset of Euclidean space can be described as the zero set of a smooth function; see for instance [6]. Now consider the system x = Gammax Gamma y xa(x; y)d; y = x Gamma y ya(x; y)d: 49) Note that the system is smooth everywhere. Let D = IR, and let A be the origin. In polar coordinates, the system (49) on IR 2 nf0g satisfies the equations r = Gammar ra(r cos ; r sin )d; ....

M. W. Hirsch, Differential Topology, Springer-Verlag, New York, 1976. 32

....ajm maths.ox. ac.uk 0 Introduction While working on complexity questions for V C dimension of general neural networks and corresponding semi Pfaffian sets [KM97a] we became convinced that major progress on o minimality should come from a more systematic use of Sard s Theorem and Morse Theory [H76]. The importance of these for Khovanski s basic work [K91] is clear. Our result was strongly confirmed by Wilkie s 1996 result [W96] on the o minimality of expansions of the real field by C 1 primitives, provided that the (necessary) condition holds that all quantifier free definable sets have ....

....and the P ij C k 2 , one also has a uniform bound. Note that we do not assume the f i are C 1 , far less real analytic. 5.4 The step from Khovanski Finiteness for systems H : IR n IR n , to the uniform bounds on connected components of zerosets, goes via Morse theory. See, for example, [H76], T86] Suppose we have functions Q i (x; f 0 (x) f k Gamma1 (x) as before, but now 1 i m say. We consider X = 1im Zer (Q i (x; f 0 (x) f k Gamma1 (x) IR n By taking the sums of the squares of the Q i , we can assume m = 1. We can also assume, at the cost of going ....

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M. W. Hirsch, Differential Topology, Springer-Verlag, 1976.

....[ i Gamma 1)m 1; im] 9) jX k (x)j g 2 for all x 2 M and all k: Since these conditions describe a Whitney C 0 open set, such vector fields exist by the lemma. The fields are bounded with respect to a complete Riemannian metric, so they have complete real analytic flows Fl Xk , see e.g. [4]. We consider the real analytic mapping f : R N M (n) f(t 1 ; t N ) 0 (Fl X1 t 1 ffi : ffi Fl XN t N ) x 1 ) Fl X1 t 1 ffi : ffi Fl XN t N ) x n ) 1 A n TRANSITIVITY OF CERTAIN DIFFEOMORPHISM GROUPS 223 which has values in the Diff (M) orbit through ....

Hirsch M. W., Differential topology, GTM 33, Springer-Verlag, New York, 1976.

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Hirsch, M.W., Differential Topology, Springer-Verlag, New York, 1976.

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M. Hirsch. Differential Topology. Springer, 1976.

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M. Hirsch. Differential Topology. Springer, 1976.

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Hirsch, M.W., Differential Topology, Springer-Verlag, New York, 1976.

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M. W. Hirsch, Differential Topology, Springer-Verlag, New York, 1976.

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Hirsch, M.W. (1976) Differential Topology, Springer Verlag, New York.

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Hirsch M. W., Differential Topology, Springer, 1976.

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Hirsch,M.W., Differential Topology, Grad. Texts Math., Vol.33, Springer-Verlag, Berlin, 1976.

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Hirsch, M. W. Differential Topology, Springer-Verlag, New York, 1976.

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