New
obstructions to the thickenig of CW complexes
Abstract
Rigidity and Glueing for Morse and
Novikov Complexes.
Abstract
We obtain rigidity and glueing results for the Morse complex of a
real-valued Morse function as well as for the Novikov complex of a
circle-valued Morse function. A rigidity result is also proved for the
Floer complex of a hamiltonian defined on a closed symplectic manifold
$(M,\omega)$ with $c_{1}|_{\pi_{2}(M)}=[\omega]|_{\pi_{2}(M)}=0$. The
rigidity results for these complexes show that the complex of a fixed
generic function/hamiltonian is a retract of the Morse (respectively
Novikov or Floer) complex of any other sufficiently $C^{0}$ close
generic function/hamiltonian. The glueing result is a type of Mayer
Vietoris formula for the Morse complex. It is used to express
algebraically the Novikov complex up to isomorphism in terms of the
Morse complex of a fundamental domain. Morse cobordisms are used to
compare various Morse-type complexes without the need of bifurcation
theory.
Homotopical Dynamics IV: Hopf
invariants and Hamiltonian flows.
Abstract
Consider
a symplectic, 1-connected manifold $M$ with a fixed riemannian metric
such that $M$ is NOT necessarily compact. For an isolated invariant set
$S$ of the negative gradient flow of a function $f$ defined on $M$ we
consider two associated invariants. The first, $c(S)$, is the Conley
index of $S$ of the negative gradient flow of $f$. The second invariant
$d(S)$ belongs to $Z/2$, is new and of a different type. The results of
the paper imply that, if $d(S)=1$, then inside any $C^{2}$
-neighbourhood of $f$ in the Whitney (strong) topology there exists a
dense family of functions $f'$ each of whose hamiltonian flow has
infinitely many distinct, periodic orbits. Even if $d(S)=0$ we may find
$f'$ as above whose hamiltonian flow has at least as many closed
(possibly homoclinic), distinct orbits as the sum of the $k$'th Betti
numbers of $c(S)$ with $k$ different from $dim(M)/2$. These results are
effective because, in many cases, $d(S)$ can be computed purely
homotopically out of an index pair $(N_{1},N_{0})$ (in the sense of the
Conley index) of $S$ (with respect to the gradient flow).
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Homotopical Dynamics III: Real
Singularities and Hamiltonian flows.
Abstract
We consider the space of "neat "singularities, that is, isolated singularities whose singular hypersurface intersects all sufficiently small spheres around the singular point transversely. On this space we discuss the stable equivalence relation induced by smooth deformations whose asymptotic behaviour is controlled by the Palais-Smale condition. It is shown that the resulting space of equivalence classes admits a canonical semi-ring structure and is isomorphic to the semi-ring of stable homotopy classes of CW-complexes.
In an
application to Hamiltonian dynamics we relate the existence of bounded
and periodic orbits on non-compact level hypersurfaces of Palais-Smale
Hamiltonians with just one singularity which is neat to the lack of
self-duality (in the sense of Spanier-Whitehead) of the sublink of the
singularity.
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Homotopical
dynamics II: Hopf invariants smoothings and the Morse complex
Abstract
We relate the connecting manifold of two consecutive critical points of a Morse-Smale function to certain Hopf invariants continuing in this fashion the classical work of John Franks. This leads to new examples of non-smoothable Poincare duality spaces and to an extension of the Morse complex.
Abstract
Flow type suspension and homotopy suspension agree for attractor-repellor homotopy data. The connection maps associated in Conley index theory to an attractor-repellor decomposition with respect to the direct flow and its inverse are Spanier-Whitehead duals in the stably parallelizable context and are duals modulo a certain Thom construction in general.
This is a short review of recent results related to the Lusternik-Schnirelmann category and functions with few critical points. The focus is on the non-Morse case.
We study the relative attaching map associated to two consecutive critical points in a context considerably more general than the Morse one. We prove that this map behaves well when suspended and that the relative attaching maps with respect to the direct and reverse flows are Spanier-Whitehead duals modulo some twisting coming from the normal structure.
We study some properties of the relative Lusternik-Schnirelmann category of Fadell. In particular, it is shown that it admits a Ganea type description. We also show that, in a compact setting, it bounds the minimal number of critical points of a certain type of functions that have a more complicated boundary behaviour than that usually allowed. This is used to relate some forms of the Arnold conjecture in symplectic geometry to the Ganea conjecture in homotopy theory.
We prove the so-called Lemaire-Sigrist conjecture for 1-connected Poincaré complexes, namely that for such spaces rational Lusternik-Schnirelmann category equals rational cone-length. We also prove that the atachment of the top cell in such a space raises rational L.-S. category by one.
Let M be a compact, smooth, 2-connected, 2m-dimensional manifold with simply connected boundary. If M has the homotopy type of an m-dimensional CW-complex, then it supports a smooth, self-indexed function, maximal, constant and regular on the boundary of Mwith at most cat(M)+2 critical points all of which are of a certain "reasonable" type. To such a critical point there corresponds, homotopically, the attachment of a cone. Conversely, to a cone attachment we may associate, under certain dimensionality and connectivity conditions, a "reasonable" critical point.
Let X be a finite type, simply connected CW-complex. If the Lusternik-Schnirelmann category of the localizations of X at each prime is bounded from above by n, then the category of X is bounded from above by 2n+1; if X is finite, this upper bound can be improved to 2n.
If a path-connected CW-complex can be covered with n+1 self-contractible, subcomplexes, then there is a homotopy equivalent space that can be constructed from a point by iteratively attaching cones, in n steps, such that, at the step k, the respective cone is attached over a k-th order suspension.
We show that the homotopic nilpotency of the algebra of piecewise polynomial forms on a simply-connected, finite type, CW-complex coincides with the strong L.S. category of the rationalization of that space. This is used to prove that, in the rational, simply-connected context all reasonable notions of cone-length agree. Both these two results are obtained as parts of a more general and functorial picture.
For a simply connected, finite type CW- complex X we introduce a geometric notion of cone-length extending the rational one introduced by Lemaire and Sigrist and we show that it is larger by at most one than the L.S. category. We also prove that the rational version agrees with the "homotopic nilpotency" of the algebra of P.L. forms on X.