Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings
Naoki Kitazawa
备注：
18 pages, 3 figures, the title changed and various errors including crucial errors in main theorems corrected, this is submitted to a (refereed academic) journal
Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions, fold maps and more general generic maps. It is one of important and interesting branches of the singularity theory of differentiable maps and applications to geometry of manifolds. In this paper we present fold maps with information of cohomology rings of their Reeb spaces. Reeb spaces are defined as the spaces of all connected components of all preimages, and in suitable situations inherit topological information such as homology groups and cohomology rings of the manifolds. Previously, the author demonstrated construction of fold maps in various cases : key methods are surgery operations to manifolds and maps and in this paper, we present more useful surgery operations and by them we construct new fold maps. More precisely, fold maps with singular value sets with crossings: the singular value set of a smooth map is the image of the set of all singular points and note that for fold maps, the set of all singular points are closed submanifolds without boundaries and the restrictions to them are immersions of codimension 1.
In this paper, as a fundamental study on the theory of Morse functions and their higher dimensional versions or fold maps and applications to geometric theory of manifolds, which were started in 1950s by differential topologists such as Thom and Whitney and have been studied actively, we study algebraic and differential topological properties of certain fold maps and their source manifolds. More precisely, we investigate fold maps obtained by surgery operations to fundamental fold maps and especially, homology groups of Reeb spaces, which are defined as the space of all connected components of…
In the singularity and differential topological theory of Morse functions and higher dimensional versions or fold maps and application to algebraic and differential topology of manifolds, constructing explicit fold maps and investigating their source manifolds is fundamental, important and difficult. The author has introduced surgery operations (bubbling operations) to fold maps, motivated by studies of Kobayashi, Saeki etc. since 1990 and has explicitly shown that homology groups of Reeb spaces of maps constructed by iterations of these operations are flexible in several cases. Such operation…
The theory of Morse functions and their higher dimensional versions or fold maps on manifolds and its application to geometric theory of manifolds is one of important branches of geometry and mathematics. Studies related to this was started in 1950s by differential topologists such as Thom and Whitney and they have been studied actively. In this paper, we study fold maps obtained by surgery operations to fundamental fold maps, and especially Reeb spaces, defined as the spaces of all connected components of preimages and in suitable situations inheriting fundamental and important algebraic inva…
We show that quasi-alternating links arise naturally when considering surgery on a strongly invertible L-space knot (that is, a knot that yields an L-space for some Dehn surgery). In particular, we show that for many known classes of L-space knots, every sufficiently large surgery may be realized as the two-fold branched cover of a quasi-alternating link. Consequently, there is considerable overlap between L-spaces obtained by surgery on $S^3$, and L-spaces resulting as two-fold branched covers of quasi-alternating links. By adapting this approach to certain Seifert fibered spaces, it is possi…
We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces to that of the base space. Finally, we show that this places restrictions on the number of finite and cyclic Dehn fillings of the knot complements in the covering spaces beyone those imposed by Culler-Gordon-Luecke-Shalen and Boyer-Zhang.
Many three dimensional manifolds are two-fold branched covers of the three dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and many examples. It shows that there are three dimensional homology spheres that do not two-fold branched cover any manifold, ones that only two-fold branched cover the three dimensional sphere, ones that just two-fold branched cover a non-trivial manifold, and ones that two-fold branched cover the sphere and non-trivial manifolds. When a manifold is surgery on a knot, the possible quotients via invo…
We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in $S^3$ in terms of homological data derived from $CFK^{\infty}(K)$. This allows us to prove some results about Dehn surgery on knots in $S^3$. In particular, we show that for a fixed manifold there are only finitely many alternating knots that can produce it by surgery. This is an improvement on a recent result by Lackenby and Purcell. We also derive a lower bound on the genus of knots depending on the…
We compute the oriented cobordism group of fold maps of 4-manifolds into the space with all the possible restrictions (and also with no restriction) to the singular fibers. We also give geometric invariants which describe completely the cobordism group of fold maps.
In this paper we develop a new theory of infinitesimal harmonic deformations for compact hyperbolic 3-manifolds with ``tubular boundary''. In particular, this applies to complements of tubes of radius at least $R_0 = \arctanh(1/\sqrt{3}) \approx 0.65848$ around the singular set of hyperbolic cone manifolds, removing the previous restrictions on cone angles. We then apply this to obtain a new quantitative version of Thurston's hyperbolic Dehn surgery theorem, showing that all generalized Dehn surgery coefficients outside a disc of ``uniform'' size yield hyperbolic structures. Here the size of a…
We give an algorithm for a surgery description of a $p$-fold cyclic branched cover of $B^3$ branched along a tangle. We generalize constructions of Montesinos and Akbulut-Kirby.
We prove that if positive integer p-surgery along a knot K \subset S^3 produces an L-space and it bounds a sharp 4-manifold, then the knot genus obeys the bound 2g(K) -1 \leq p - \sqrt{3p+1}. Moreover, there exists an infinite family of pairs (K_n,p_n) attaining this bound, where K_n denotes an n-fold iterated cable of the unknot and p_n \to \infty. In particular, the stated bound applies when the knot surgery produces a lens space or a connected sum thereof. Combined with work of Gordon-Luecke, Hoffman, and Matignon-Sayari, it follows that if surgery along a knot produces a connected sum of l…
The inertia subgroup $I_n(π)$ of a surgery obstruction group $L_n(π)$ is generated by elements which act trivially on the set of homotopy triangulations $\Cal S(X)$ for some closed topological manifold $X^{n-1}$ with $π_1(X)=π$. This group is a subgroup of the group $C_n(π)$ which consists of the elements which can be realized by normal maps of closed manifolds. In all known cases these groups coincide and the computation of them is one of the basic problems of surgery theory. The computation of the group $C_n(π)$ is equivalent to the computation the image of the assembly map $A:H_{n}(Bπ, \bol…
Fold maps are higher dimensional versions of Morse functions and fundamental and important tools in studying algebraic and differential topological properties of manifolds: as the theory established by Morse and the higher dimensional version, started by Thom and Whitney, later actively studied by Eliashberg, Levine etc. and recently studied by Kobayashi, Saeki, Sakuma etc., explicitly show this. One of fundamental, important and difficult studies on this field is, constructing explicit fold maps and investigating their source manifolds. As fundamental and strong tools for systematic construct…
Examples suggest that there is a correspondence between L-spaces and 3-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of such manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric 3-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the 2-fold branched covers of non-split alternating links. To do this we prove that the fundamental group …
