Surgery on links with unknotted components and three-manifolds

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link L with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on the same link L could give the same three-manifold M.Comment: 10 pages, 8 figure

What is NP? - Interpretation of a Chinese paradox "white horse is not horse"

The notion of nondeterminism has disappeared from the current definition of NP, which has led to ambiguities in understanding NP, and caused fundamental difficulties in studying the relation P versus NP. In this paper, we question the equivalence of the two definitions of NP, the one defining NP as the class of problems solvable by a nondeterministic Turing machine in polynomial time, and the other defining NP as the class of problems verifiable by a deterministic Turing machine in polynomial time, and reveal cognitive biases in this equivalence. Inspired from a famous Chinese paradox white horse is not horse, we further analyze these cognitive biases. The work shows that these cognitive biases arise from the confusion between different levels of nondeterminism and determinism, due to the lack of understanding about the essence of nondeterminism. Therefore, we argue that fundamental difficulties in understanding P versus NP lie firstly at cognition level, then logic level

On free ZpZ_p-torus actions in dimension two and three

We confirm the Halperin-Carlsson Conjecture for free $Z_p$-torus actions (p is a prime) on 2-dimensional finite CW-complexes and free $Z_2$-torus actions on compact 3-manifolds.Comment: 26 pages, no figure. The contents of the paper are reorganized and some proofs are simplifie

On lower bounds of the sum of multigraded Betti numbers of simplicial complexes

We find some general lower bounds of the sum of certain families of multigraded Betti numbers of any simplicial complex with a vertex coloring.Comment: 15 pages, 2 figures. Minor revisions are made (two pictures and some new references are added

On the constructions of free and locally standard Z_2-torus actions on Manifolds

We introduce an elementary way of constructing principal (Z_2)^m-bundles over compact smooth manifolds. In addition, we will define a general notion of locally standard (Z_2)^m-actions on closed manifolds for all m>0, and then give a general way to construct all such (Z_2)^m-actions from the orbit space. Some related topology problems are also studied.Comment: 28 pages, 12 figures, some minor revisions are made, one picture and one reference are added

Quantum Boson Algebra and Poisson Geometry of the Flag Variety

In his work on crystal bases \cite{Kas}, Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semi-simple Lie algebra $\mathfrak g$, which he called a quantum boson algebra. In this paper, we construct Kashiwara operators associated to all positive roots and use them to define a variant of Kashiwara's quantum boson algebra. We show that a quasi-classical limit of the positive half of our variant is a Poisson algebra of the form $(P \simeq \mathbb C[\mathfrak n^{\ast}], \, \{~~,~~\}_P)$, where $\mathfrak n$ is the positive part of $\mathfrak g$ and $\{~~,~~\}_P$ is a Poisson bracket that has the same rank as, but is different from, the Kirillov-Kostant bracket $\{~~,~~\}_{KK}$ on $\mathfrak n^{\ast}$. Furthermore, we prove that, in the special case of type $A$, any linear combination $a_1 \{~~,~~\}_P + a_2 \{~~,~~\}_{KK}$, $a_1, a_2 \in \mathbb C$, is again a Poisson bracket. In the general case, we establish an isomorphism of $P$ and the Poisson algebra of regular functions on the open Bruhat cell in the flag variety. In type $A$, we also construct a Casimir function on the open Bruhat cell, together with its quantization, which may be thought of as an analog of the linear function on $\mathfrak n^{\ast}$ defined by a root vector for the highest root

Cubes and Generalized Real Bott Manifolds

We define a notion of facets-pairing structure and its seal space on a nice manifold with corners. We will study facets-pairing structures on any cube in detail and investigate when the seal space of a facets-pairing structure on a cube is a closed manifold. In particular, for any binary square matrix $A$ with zero diagonal in dimension n, there is a canonical facets-pairing structure $F_A$ on the n-dimensional cube. We will show that all the closed manifolds that we can obtain from the seal spaces of such $F_A$'s are neither more nor less than all the generalized real Bott manifolds --- a special class of real toric manifolds introduced by Choi, Masuda and Suh.Comment: Some small changes were made to the previous version. The introduction part was expanded and a new reference was adde

Gaiotto's Lagrangian subvarieties via loop groups

The purpose of this note is to give a simple proof of the fact that a certain substack, defined in [2], of the moduli stack $T^{\ast}Bun_G(\Sigma)$ of Higgs bundles over a curve $\Sigma$, for a connected, simply connected semisimple group $G$, possesses a Lagrangian structure. The substack, roughly speaking, consists of images under the moment map of global sections of principal $G$-bundles over $\Sigma$ twisted by a smooth symplectic variety with a Hamiltonian $G$-action

On Hochster's formula for a class of quotient spaces of moment-angle complexes

Any finite simplicial complex K and a partition of the vertex set of K determines a canonical quotient space of the moment-angle complex of K. We prove that the cohomology groups of such a space can be computed via some Hochster's type formula, which generalizes the usual Hochster's formula for the cohomology groups of moment-angle complexes. In addition, we show that the stable decomposition of moment-angle complexes can also be extended to such spaces. This type of spaces include all the quasitoric manifolds that are pullback from the linear models. And we prove that the moment-angle complex associated to a finite simplicial poset is always homotopy equivalent to one of such spaces.Comment: 19 pages, 3 figures. The paper is significantly simplified from the previous version while retains all the main result

Case Study of the Proof of Cook's theorem - Interpretation of A(w)

Cook's theorem is commonly expressed such as any polynomial time-verifiable problem can be reduced to the SAT problem. The proof of Cook's theorem consists in constructing a propositional formula A(w) to simulate a computation of TM, and such A(w) is claimed to be CNF to represent a polynomial time-verifiable problem w. In this paper, we investigate A(w) through a very simple example and show that, A(w) has just an appearance of CNF, but not a true logical form. This case study suggests that there exists the begging the question in Cook's theorem