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Register a new userOn the Browder-Levine-Novikov embedding theorems
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In this survey we present applications of the ideas of complement and neighborhood in the theory embeddings of manifolds into Euclidean space (in codimension at least three). We describe how the combination of these ideas gives a reduction of embeddability and isotopy problems to algebraic problems. We present a more clarified exposition of the Browder-Levine theorem on realization of normal systems. Most of the survey is accessible to non-specialists in the theory of embeddings.
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A Banach space operator $Tin B({cal X})$ is polaroid if points $lambdainisosigmasigma(T)$ are poles of the resolvent of $T$. Let $sigma_a(T)$, $sigma_w(T)$, $sigma_{aw}(T)$, $sigma_{SF_+}(T)$ and $sigma_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $Cin B({cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $pi_0(M_C)={lambdainisosigma(M_C) 0<dim(M_C-lambda)^{-1}(0)<infty}$, $M_0$ satisfies Weyls theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $lambdainsigma_w(M_0)setminussigma_{SF_+}(A)$ and $B$ has SVEP at points $muinsigma_w(M_0)setminussigma_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $lambdainsigma_w(M_0)setminussigma_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $lambdainsigma_w(M_0)setminussigma_{SF_+}(A)$ and $B^*$ has SVEP at points $muinsigma_w(M_0)setminussigma_{SF_-}(B)$, then $sigma(M_C)setminussigma_w(M_C)=pi_0(M_C)$. Here the hypothesis that $lambdainpi_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $lambdainpi_0(A)$ are poles of the resolvent of $A$. For an operator $Tin B(X)$, let $pi_0^a(T)={lambda:lambdainisosigma_a(T), 0<dim(T-lambda)^{-1}(0)<infty}$. We prove that if $A^*$ and $B^*$ have SVEP, $A$ is polaroid on $pi_0^a(M)$ and $B$ is polaroid on $pi_0^a(B)$, then $sigma_a(M)setminussigma_{aw}(M)=pi_0^a(M)$.
For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this formula in terms of the square of the linking number and the second coefficient of the Conway polynomial.
An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_e(G)$ of $(G, d)$ is the lowest genus of a surface on which such an embedding is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $ggeq g_e(G)$, $(G, d)$ admits such an embedding (possibly after a rescaling of $d$) on a surface of genus $g$. Next, we study minimal embeddings where each complementary region has Euler characteristic $-1$. The maximum essential genus $g_e^{max}(G)$ of $(G, d)$ is the largest genus of a surface on which the graph is minimally embedded. Finally, we describe a method explicitly for an essential embedding of $(G, d)$, where $g_e(G)$ and $g_e^{max}(G)$ are realized.
In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, the second author gave integral lifts of the Conway-Gordon theorems in terms of the square of the linking number and the second coefficient of the Conway polynomial. In this paper, we generalize the integral Conway-Gordon theorems to complete graphs with arbitrary number of vertices greater than or equal to six. As an application, we show that for every rectilinear spatial complete graph whose number of vertices is greater than or equal to six, the sum of the second coefficients of the Conway polynomials over all of the Hamiltonian knots is determined explicitly in terms of the number of triangle-triangle Hopf links.
Conway-Gordon proved that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this paper, we give a Conway-Gordon type theorem for any graph which is obtained from the complete graph on 6 or 7 vertices by a finite sequence of $triangle Y$-exchanges.
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