On the h-Cobordism Category I

2019 ◽  
Author(s):  
George Raptis ◽  
Wolfgang Steimle
Keyword(s):  
Homotopy Type ◽  
Smooth Manifold ◽  
Manifolds With Boundary ◽  
Smooth Manifolds ◽  
Topological Category

Abstract We consider the topological category of $h$-cobordisms between smooth manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.

How Likely is the Variety of Singularities of a Serial Manipulator a Smooth Manifold?

2010 ◽  
Author(s):  
Andreas Mu¨ller
Keyword(s):  
Smooth Manifold ◽  
Sufficient Condition ◽  
Smooth Manifolds ◽  
Generic Properties ◽  
End Effector ◽  
Serial Manipulators ◽  
The Individual ◽  
Forward Kinematic

It is commonly assumed that the singularities of serial manpulators constitute generically smooth manifolds. This assumption has not been proved yet, nor is there an established notion of of genericity. In this paper two different notions of generic properties of forward kinematic mappings are discussed. Serial manipulators are classified according to the type of end-effector motion and feasible manipulator geometries. A sufficient condition for that generically singularities form locally smooth manifolds is presented. This condition admits to separately treat the individual classes. As an example it is shown that the singularities of 3-DOF manipulators form generically smooth manifolds.

Foldings of star manifolds

1993 ◽  
Vol 123 (6) ◽  
pp. 1011-1016
Author(s):  
H. R. Farran ◽  
E. El-Kholy ◽  
S. A. Robertson
Keyword(s):  
Smooth Manifold ◽  
General Setting ◽  
Smooth Manifolds ◽  
Inductive Definition ◽  
Metrical Structure ◽  
Totally Geodesic Submanifold ◽  
Geodesic Submanifold ◽  
Crucial Feature ◽  
Definition Of

SynopsisThis paper is a sequel to [4]. Its purpose is to show that the concept of isometric foldings of Riemannian manifolds can be extended to a much wider class of manifolds without losing the main structure theorem. We present here what we believe to be a definitive form of the folding concept for smooth manifolds.The theory discussed here is based on the idea of a 1-spread [2], where the role of geodesies on a Riemannian manifold is assumed by smooth, unoriented and unparametrised curves on a smooth manifold. The absence of metrical structure forces a fresh approach to the basic definitions. A crucial feature of the Riemannian theory does survive, however, in this general setting: a 1-spread on a sufficiently smooth manifold M induces a 1-spread on sufficiently small spheres surrounding any point of M. With the help of this fact, we are able to construct an inductive definition of “star folding” f:M → N between smooth manifolds M and N, and to retain the theorem that the manifold M is stratified by the “folds”, each of which has the character of a “totally geodesic” submanifold with respect to the above-mentioned curves.

On the ARF Invariant of an Involution

1970 ◽  
Vol 22 (3) ◽  
pp. 519-524 ◽  
Author(s):  
Peter Orlik
Keyword(s):  
Abelian Group ◽  
Smooth Manifold ◽  
Finite Abelian Group ◽  
Homotopy Equivalent ◽  
Smooth Manifolds ◽  
Connected Sum ◽  
Group Operation ◽  
Free Involution ◽  
Arf Invariant

Let Σ4k+1 denote a smooth manifold homeomorphic to the (4k + 1)-sphere, S4k+1k ≧ 1, and T: Σ4k+1 → Σ4k+1 a differentiate free involution. Our aim in this note is to derive a connection between the differentiate structure on Σ4k+1 and the properties of the free involution T.To be more specific, recall [5] that the h-cobordism classes of smooth manifolds homeomorphic (or, what is the same, homotopy equivalent) to S4k+1, k ≧ 1, form a finite abelian group θ4k+1 with group operation connected sum. The elements are called homotopy spheres. Those homotopy spheres that bound parallelizable manifolds form a subg roup bP4k+2 ⊂ θ4k+1. It is proved in [5, Theorem 8.5] that bP4k+2 is either zero or cyclic of order 2. In the latter case the two distinct homotopy spheres are distinguished by the Arf invariant of the parallelizable manifolds they bound.

scholarly journals Manifolds of mappings on Cartesian products

2022 ◽  
Author(s):  
Helge Glöckner ◽  
Alexander Schmeding
Keyword(s):  
Smooth Manifold ◽  
Directional Derivatives ◽  
Locally Convex Spaces ◽  
Smooth Manifolds ◽  
Cartesian Products ◽  
Manifold Structure ◽  
Derivatives Of ◽  
Locally Convex ◽  
Convex Spaces

AbstractGiven smooth manifolds $$M_1,\ldots , M_n$$ M 1 , … , M n (which may have a boundary or corners), a smooth manifold N modeled on locally convex spaces and $$\alpha \in ({{\mathbb {N}}}_0\cup \{\infty \})^n$$ α ∈ ( N 0 ∪ { ∞ } ) n , we consider the set $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) of all mappings $$f:M_1\times \cdots \times M_n\rightarrow N$$ f : M 1 × ⋯ × M n → N which are $$C^\alpha $$ C α in the sense of Alzaareer. Such mappings admit, simultaneously, continuous iterated directional derivatives of orders $$\le \alpha _j$$ ≤ α j in the jth variable for $$j\in \{1,\ldots , n\}$$ j ∈ { 1 , … , n } , in local charts. We show that $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) admits a canonical smooth manifold structure whenever each $$M_j$$ M j is compact and N admits a local addition. The case of non-compact domains is also considered.

Diffeomorphisms of some smooth metastably connected manifolds

1972 ◽  
Vol 71 (1) ◽  
pp. 19-23 ◽  
Author(s):  
J. P. E. Hodgson
Keyword(s):  
Exact Sequence ◽  
Smooth Manifold ◽  
Smooth Manifolds ◽  
Alternative Description ◽  
Main Portion ◽  
Image Position

Let Mm be a closed connected smooth manifold of dimension m, and set Pm = Mm – int Dm where Dm is a disc in M. In (4), Wall has the following exact sequencewhere (M), (resp. (P)) is the Δ-set of diffeomorphisms of M (resp. P) as in(l), and [M/P] is the set of diffeomorphism classes of smooth manifolds obtained by glueing a disc to the boundary of P. In this paper we obtain some results on π0((M)) for particular M, and in the following sense: Using the techniques of (2), we can determine π0((P)), so the main portion of the paper is concerned with a discussion of the kernel of α. There is a map ω: π1((P)) → Γm+1, given by the obstruction to extending h∈π1((P)) to a concordance of the identity of M to itself, and it is clear that if this map is attached at the beginning of the sequence (*) we get exactness. We will obtain (for certain M) an alternative description of the image of α in terms of those homotopy spheres which can appear as the boundaries of thickenings.

scholarly journals Chaotic diffeomorphism on manifold (smooth manifold)

2019 ◽  
Vol 24 (1) ◽  
pp. 103
Author(s):  
Ali A. Shihab1 ◽  
Mizal H. Al-Obaidi2 ◽  
Jehan Hamad Hazaa2
Keyword(s):  
Smooth Manifold ◽  
Chaotic Maps ◽  
Smooth Manifolds ◽  
Dense Product

In this paper  we concern in studying chaotic homeomorphisms deals with study and investigate of chaotic homeomorphisms on smooth manifolds. For connections more precisely to chaotic diffeomorphisms maps on smooth manifolds. New relation between diffeomorphism an chaotic, transitive, dense and nowhere dense in smooth manifolds have been found. Product of two diffeomorphisms dense maps on smooth manifold is dense. Product of two chaotic maps on smooth manifold is chaotic.   http://dx.doi.org/10.25130/tjps.24.2019.017

scholarly journals Geometry of the free-sliding Bernoulli beam

2016 ◽  
Vol 24 (2) ◽  
pp. 153-171
Author(s):  
Giovanni Moreno ◽  
Monika Ewa Stypa
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Variational Problem ◽  
Manifolds With Boundary ◽  
Boundary Values ◽  
Smooth Manifolds ◽  
Bernoulli Beam ◽  
Variation Process

Abstract If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of the free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed curves. There exists a rigorous geometric way to formulate this sort of problems on smooth manifolds with boundary, which we review here in a friendly self-contained way. As an application, we study the particular free boundary values variational problem of the free-sliding Bernoulli beam.

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