Smooth Homotopy Research Articles

Algebraic periods and minimal number of periodic points for smooth self-maps of 1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{1}$$\\end{document}-connected 4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{4}$$\\end{document}-manifolds with definite intersection forms

Let M be a closed 1-connected smooth 4-manifolds, and let r be a non-negative integer. We study the problem of finding minimal number of r-periodic points in the smooth homotopy class of a given map f:M→M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:M \\rightarrow M$$\\end{document}. This task is related to determining a topological invariant Dr4[f]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^4_r[f]$$\\end{document}, defined in Graff and Jezierski (Forum Math 21(3):491–509, 2009), expressed in terms of Lefschetz numbers of iterations and local fixed point indices of iterations. Previously, the invariant was computed for self-maps of some 3-manifolds. In this paper, we compute the invariants Dr4[f]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^4_r[f]$$\\end{document} for the self-maps of closed 1-connected smooth 4-manifolds with definite intersection forms (i.e., connected sums of complex projective planes). We also present some efficient algorithmic approach to investigate that problem

Ribbonness of Kervaire’s Sphere-Link in Homotopy 4-Sphere and its Consequences to 2-Complexes

M. A. Kervaire showed that every group of deficiency d and weight d is the fundamental group of a smooth sphere-link of d components in a smooth homotopy 4-sphere. In the use of the smooth unknotting conjecture and the smooth 4D Poincar´e conjecture, any such sphere-link is shown to be a sublink of a free ribbon sphere-link in the 4-sphere. Since every ribbon sphere-link in the 4-sphere is also shown to be a sublink of a free ribbon sphere-link in the 4-sphere, Kervaire’s sphere-link and the ribbon sphere-link are equivalent con- cepts. By applying this result to a ribbon disklink in the 4-disk, it is shown that the compact complement of every ribbon disk-link in the 4-disk is aspherical. By this property, a ribbon disk-link presentation for every contractible finite 2-complex is introduced. By using this presentation, it is shown that every connected subcomplex of a contractible finite 2-complex is aspherical (meaning partially yes for Whitehead aspherical conjecture).

C∞-manifolds with skeletal diffeology

We formulate and study the notion of d-skeletal diffeology, which generalizes that of wire diffeology, introducing the dual notion of d-coskeletal diffeology. We first show that paracompact finite-dimensional C∞-manifolds Md with d-skeletal diffeology inherit good topologies and smooth paracompactness from M. We then study the pathology of Md. Above all, we prove the following: For d<dimM, every immersion f:M⟶N is isolated in the diffeological space D(Md,Nd) of smooth maps and the d-dimensional smooth homotopy group of Md is uncountable.

Smooth Homotopy of Infinite-Dimensional 𝐶^{∞}-Manifolds

In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional C ∞ C^{\infty } -manifolds in convenient calculus. More precisely, we discuss the smoothing of maps, sections, principal bundles, and gauge transformations. We first introduce the notion of hereditary C ∞ C^\infty -paracompactness along with the semiclassicality condition on a C ∞ C^\infty -manifold, which enables us to use local convexity in local arguments. Then, we prove that for C ∞ C^\infty -manifolds M M and N N , the smooth singular complex of the diffeological space C ∞ ( M , N ) C^\infty (M,N) is weakly equivalent to the ordinary singular complex of the topological space C 0 ( M , N ) {\mathcal {C}^0}(M,N) under the hereditary C ∞ C^\infty -paracompactness and semiclassicality conditions on M M . We next generalize this result to sections of fiber bundles over a C ∞ C^\infty -manifold M M under the same conditions on M M . Further, we establish the Dwyer-Kan equivalence between the simplicial groupoid of smooth principal G G -bundles over M M and that of continuous principal G G -bundles over M M for a Lie group G G and a C ∞ C^\infty -manifold M M under the same conditions on M M , encoding the smoothing results for principal bundles and gauge transformations. For the proofs, we fully faithfully embed the category C ∞ C^{\infty } of C ∞ C^{\infty } -manifolds into the category D {\mathcal {D}} of diffeological spaces and develop the smooth homotopy theory of diffeological spaces via a homotopical algebraic study of the model category D {\mathcal {D}} and the model category C 0 {\mathcal {C}^0} of arc-generated spaces, also known as Δ \Delta -generated spaces. Then, the hereditary C ∞ C^\infty -paracompactness and semiclassicality conditions on M M imply that M M has the smooth homotopy type of a cofibrant object in D {\mathcal {D}} . This result can be regarded as a smooth refinement of the results of Milnor, Palais, and Heisey, which give sufficient conditions under which an infinite-dimensional topological manifold has the homotopy type of a C W CW -complex. We also show that most of the important C ∞ C^\infty -manifolds introduced and studied by Kriegl, Michor, and their coauthors are hereditarily C ∞ C^\infty -paracompact and semiclassical, and hence, results can be applied to them.

Periodic points of self-maps of products of lens spaces $L(3)\times L(3)$

Let $f\colon M\to M$ be a self-map of a compact manifold and $n\in \mathbb{N}$. The least number of $n$-periodic points in the smooth homotopy class of $f$ may be smaller than in the continuous homotopy class. We ask: for which self-maps $f\colon M\to M$ the two minima are the same, for each prescribed multiplicity? In the study of self-maps of tori and compact Lie groups a necessary condition appears. Here we give a criterion which helps to decide whether the necessary condition is also sufficient. We apply this result to show that for self-maps of the product of the lens space $M=L(3)\times L(3)$ the necessary condition is also sufficient.

Gluck twisting roll spun knots

We show that the smooth homotopy 4-sphere obtained by Gluck twisting the m-twist n-roll spin of any unknotting number one knot is diffeomorphic to the standard 4-sphere, for any pair of integers (m,n). It follows as a corollary that an infinite collection of twisted doubles of Gompf's infinite order corks are standard.

Least number of n-periodic points of self-maps of PSU(2)times PSU(2)

Let f:Mrightarrow M be a self-map of a compact manifold and nin {mathbb {N}}. In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration f^n, appears. Here we give the explicit form of the graph of orbits of Reidemeister classes mathcal {GOR}(f^*) for self-maps of projective unitary group PSU(2) and of PSU(2)times PSU(2) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations.

An aggregate homotopy method for solving unconstrained minimax problems

ABSTRACT In this paper, an aggregate homotopy method is proposed to solve unconstrained minimax problems. The homotopy mapping is constructed by the linear homotopy and the aggregate function for the objective function, and the homotopy parameter is also used as the smoothing parameter of the aggregate function. Under some general assumptions, the existence and global convergence of a smooth homotopy path are proved for almost all starting points in or a ball set, and a stationary point of the unconstrained minimax problem can be obtained. A path-following procedure is introduced to numerically trace the homotopy path. When the smoothing parameter becomes small enough, an alternative strategy of the path-following procedure is given to reduce the ill-conditioning of the aggregate function, it uses the Newton method to solve a nonlinear system, which is derived from the first order optimality conditions for the unconstrained minimax problem. Numerical results show that the proposed method is efficient and robust.

An A∞ version of the Poincaré lemma

We prove a categorified version of the Poincar\'e lemma. The natural setting for our result is that of $\infty$-local systems. More precisely, we show that any smooth homotopy between maps $f$ and $g$ induces an $\mathsf{A}_\infty$-natural transformation between the corresponding pullback functors. This transformation is explicitly defined in terms of Chen's iterated integrals. In particular, we show that a homotopy equivalence induces a quasi-equivalence on the DG categories of $\infty$-local system.

Removing isolated zeroes by homotopy

Suppose that the inverse image of the zero vector by a continuous map $f:{\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. There is a local obstruction to removing this isolated zero by a small perturbation, generalizing the notion of index for vector fields, the $q=n$ case. The existence of a continuous map $g$ which approximates $f$ but is nonvanishing near $P$ is equivalent to a topological property we call "locally inessential," and for dimensions $n$, $q$ where $\pi_{n-1}(S^{q-1})$ is trivial, every isolated zero is locally inessential. We consider the problem of constructing such an approximation $g$, and show that there exists a continuous homotopy from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic map, then there exists such a homotopy which is also semialgebraic. For $q=2$ and $f$ real analytic with a locally inessential isolated zero, there exists a H\"older continuous homotopy $F(x,t)$ which, for $(x,t)\ne(P,0)$, is real analytic and nonvanishing. The existence of a smooth homotopy, given a smooth map $f$, is stated as an open question.

Model category of diffeological spaces

The existence of a model structure on the category \({\mathcal {D}}\) of diffeological spaces is crucial to developing smooth homotopy theory. We construct a compactly generated model structure on the category \({\mathcal {D}}\) whose weak equivalences are just smooth maps inducing isomorphisms on smooth homotopy groups. The essential part of our construction of the model structure on \({\mathcal {D}}\) is to introduce diffeologies on the sets \(\varDelta ^{p}\)\((p \ge 0)\) such that \(\varDelta ^{p}\) contains the \(k\mathrm{th}\) horn \(\varLambda ^{p}_{k}\) as a smooth deformation retract.

A smooth path-following algorithm for market equilibrium under a class of piecewise-smooth concave utilities

This paper presents a smooth path-following algorithm for computing market equilibrium in a pure exchange economy under a class of piecewise-smooth concave utilities, which can be expressed as $$u(x)=\min _\ell \{f_\ell (x)\}$$ with $$f_\ell (x)$$ being a smooth concave function for all $$\ell $$ . As a result of a smooth technique for minimax problems, a smooth homotopy mapping is derived from the introduction of logarithmic barrier terms and an extra variable. With this mapping, it is proved that there always exists a smooth path leading to a market equilibrium as the extra variable approaches zero. A predictor–corrector method is adapted for numerically following this path. Numerical results are given to further demonstrate the effectiveness and efficiency of the algorithm.

A Smooth Homotopy Method for the General Equilibrium Model With Incomplete Asset Markets

A Smooth Homotopy Method for the General Equilibrium Model With Incomplete Asset Markets

Fuzzy smooth homotopy on fuzzy Banach manifold

In this paper the structure of fuzzy Banach manifolds has been enriched by inducing three different equivalence relations on fuzzy Banach atlases. Also a Network fuzzy Banach manifold has been defined admitting two equivalence relation and two group structures. Further we define fuzzy smooth homotopy on fuzzy Banach manifold and the study has been extended for three different fuzzy path connectedness inducing equivalence relations and fundamental group structure which is invariant under fuzzy homeomorphism.

Real projective spaces with all geodesics closed

Let M be a smooth n-manifold admitting an even degree smooth covering by a smooth homotopy n-sphere. When \({n \neq 3}\), we prove that if g is a Riemannian metric on M with all geodesics closed, then g has constant sectional curvatures. In particular, n-dimensional real projective spaces (\({n \neq 3}\)) with all geodesics closed have constant sectional curvatures.

Smooth structures on a fake real projective space

We show that the group of smooth homotopy 7-spheres acts freely on the set of smooth manifold structures on a topological manifold M which is homotopy equivalent to the real projective 7-space. We classify, up to diffeomorphism, all closed manifolds homeomorphic to the real projective 7-space. We also show that M has, up to diffeomorphism, exactly 28 distinct differentiable structures with the same underlying PL structure of M and 56 distinct differentiable structures with the same underlying topological structure of M.

INERTIA GROUPS AND SMOOTH STRUCTURES OF (n-1)-CONNECTED 2n-MANIFOLDS

Let $M^{2n}$ denote a closed ($n-1$)-connected smoothable topological $2n$-manifold. We show that the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ is isomorphic to the group of smooth homotopy spheres $\bar{\Theta}_{2n}$ for $n=4$ or $5$, the concordance inertia group $I_{c}(M^{2n})=0$ for $n=3$, $4$, $5$ or $11$ and the homotopy inertia group $I_{h}(M^{2n})=0$ for $n=4$. On the way, following Wall's approach [16] we present a new proof of the main result in [9], namely, for $n=4$, $8$ and $H^{n}(M^{2n};\mathbb{Z})\cong \mathbb{Z}$, the inertia group $I(M^{2n})\cong \mathbb{Z}_{2}$. We also show that, up to orientation-preserving diffeomorphism, $M^{8}$ has at most two distinct smooth structures; $M^{10}$ has exactly six distinct smooth structures and then show that if $M^{14}$ is a $\pi$-manifold, $M^{14}$ has exactly two distinct smooth structures.

Homotopy method for a class of multiobjective optimization problems with equilibrium constraints

In this paper, we present a combined homotopy interior point method for solving multiobjective programs with equilibrium constraints. Under suitable conditions, we prove the existence and convergence of a smooth homotopy path from almost any interior point to a solution of the K-K-T system. Numerical results are presented to show the effectiveness of this algorithm.

Bergman-harmonic maps of balls

We study Bergman-harmonic maps between balls 8 : Bn ! BN extending of class either C2 orM1 to the boundary of Bn. For every holomorphic (anti-holomorphic) map 8 : Bn ! BN extending smoothly to the boundary and every smooth homotopy H : 8 ' 9 we prove a Lichnerowicz-type (cf. [28]) result, i.e., we show that E✏ (9) # E✏ (8) + O(✏−n+1). When 8 is proper, Bergman-harmonic, and C2 up to the boundary, the boundary values map % : S2n−1 ! S2N−1 is shown to satisfy a compatibility system similar to the tangential Cauchy-Riemann equations on S2n−1 (and satisfied by the boundary values of any proper holomorphic map). For every weakly Bergman-harmonic map 8 2 W1(Bn,BN ) admitting Sobolev boundary values % 2 M1(S2n−1,BN ) in the sense of [6], the boundary values % are shown to be a weakly subelliptic harmonic map of (S2n−1, ⌘) into (BN , h), provided that 8−1rh stays bounded at the boundary of Bn and % has vanishing weak normal derivatives.

An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds

For a given self-map $f$ of $M$, a closed smooth connected and simply-connected manifold of dimension $m\geq 4$, we provide an algorithm for estimating the values of the topological invariant $D^m_r[f]$, which equals the minimal number of $r$-periodic points in the smooth homotopy class of $f$. Our results are based on the combinatorial scheme for computing $D^m_r[f]$ introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013), 63-84]. An open-source implementation of the algorithm programmed in C++ is publicly available at {\tt http://www.pawelpilarczyk.com/combtop/}.