Results 11  20
of
43
Morse functions and cohomology of homogeneous spaces
, 2004
"... Morse functions are useful tool in revealing the geometric formation of its domain manifolds M. They define the handle decompositions of M from which the additive homologies H∗(M) may be constructed. In these lectures two further questions were emphasized. (1) How to find a Morse function on a given ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Morse functions are useful tool in revealing the geometric formation of its domain manifolds M. They define the handle decompositions of M from which the additive homologies H∗(M) may be constructed. In these lectures two further questions were emphasized. (1) How to find a Morse function on a given manifold? (2) From Morse functions can one derive the multiplicative cohomology rather than the additive homology? It is not our intention here to make detailed studies of these question. Instead, we will illustrate by examples solutions to them for some classical manifolds as homogeneous spaces.
Computing Cocycles on Simplicial Complexes
"... In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20,21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20,21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we give here an algorithm for computing cupi products over integers on a simplicial complex at chain level. 1
POINCARÉ DUALITY SPACES
"... At the end of the last century, Poincaré discovered that the Betti numbers of a closed oriented triangulated topological nmanifold X n satisfy the relation bi(X): = dimRHi(X; R) ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
At the end of the last century, Poincaré discovered that the Betti numbers of a closed oriented triangulated topological nmanifold X n satisfy the relation bi(X): = dimRHi(X; R)
A Note About Universality Theorem as an Enumerative RiemannRoch Theorem, preprint AG/0405113
, 2004
"... This short note is a supplement of the longer paper [Liu6], in which the author gives an algebraic proof of the following universality theorem. Theorem 1 Let δ ∈ N denote the number of nodal singularities. Let L be a 5δ − 1 veryample 1 line bundle on an algebraic surface M, then the number of δ−nod ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
This short note is a supplement of the longer paper [Liu6], in which the author gives an algebraic proof of the following universality theorem. Theorem 1 Let δ ∈ N denote the number of nodal singularities. Let L be a 5δ − 1 veryample 1 line bundle on an algebraic surface M, then the number of δ−nodes nodal singular curves in a generic δ dimensional linear subsystem of L  can be expressed as a universal polynomial (independent to M) of c1(L) 2, c1(L) · c1(M), c1(M) 2, c2(M) of degree δ. The finiteness of the “number of δ−nodes nodal singular curves in a generic δ dimensional linear subsystem of L ” was proved by Göttsche in [Got] proposition 5.2. Our theorem shows that these numbers are topological invariants of (M, L). A weaker form of the above statement has appeared in [Got] based on results in [V] and [KP] for small δ. In his conjecture, Göttsche had assumed the existence of a lower bound m0, such that for any very ample L0, the statement in the above theorem holds for L = L ⊗m 0, with m ≥ m0. The purpose of the note is to address upon the geometric and topological meanings of the universality theorem and its relationship with the well known surface RiemannRoch formula of algebraic
The Incommunicability of Content
 Mind
, 1966
"... 1. Setting up the foundations 3 2. The EilenbergSteenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
1. Setting up the foundations 3 2. The EilenbergSteenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6
Nielsen numbers of nvalued fiber maps
 Journal of Fixed Point Theory and Applications
, 2008
"... The Nielsen number for nvalued multimaps, defined by Schirmer, has been calculated only for the circle. A concept of nvalued fiber map on the total space of a fibration is introduced. A formula for the Nielsen numbers of nvalued fiber maps of fibrations over the circle reduces the calculation to ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
The Nielsen number for nvalued multimaps, defined by Schirmer, has been calculated only for the circle. A concept of nvalued fiber map on the total space of a fibration is introduced. A formula for the Nielsen numbers of nvalued fiber maps of fibrations over the circle reduces the calculation to the computation of Nielsen numbers of singlevalued maps. If the fibration is orientable, the product formula for singlevalued fiber maps fails to generalize, but a “semiproduct formula ” is obtained. In this way, the class of nvalued multimaps for which the Nielsen number can be computed is substantially enlarged. Subject Classification 55M20, 54C60 1
On higher dimensional HirzebruchJung singularities
 Rev. Mat. Complut
"... A germ of normal complex analytical surface is called a HirzebruchJung singularity if it is analytically isomorphic to the germ at the 0dimensional orbit of an affine toric surface. Two such germs are known to be isomorphic if and only if the toric surfaces corresponding to them are equivariantly ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
A germ of normal complex analytical surface is called a HirzebruchJung singularity if it is analytically isomorphic to the germ at the 0dimensional orbit of an affine toric surface. Two such germs are known to be isomorphic if and only if the toric surfaces corresponding to them are equivariantly isomorphic. We extend this result to higherdimensional HirzebruchJung singularities, which we define to be the germs analytically isomorphic to the germ at the 0dimensional orbit of an affine toric variety determined by a lattice and a simplicial cone of maximal dimension. We deduce a normalization algorithm for quasiordinary hypersurface singularities. 2000 Mathematics Subject Classification. Primary 32S10; Secondary 14M25. 1
A History of Duality in Algebraic Topology
"... This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R
Topology, matter and space, i: Topological notions in 19thcentury natural philosophy. Archive for History of Exact Sciences 52
, 1998
"... I. Topological notions in the weave of scientific practice......................................................299 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
I. Topological notions in the weave of scientific practice......................................................299
Applied Categories and Functors for Undergraduates
, 810
"... These are lecture notes for a 1–semester undergraduate course (in computer science, mathematics, physics, engineering, chemistry or biology) in applied categorical metalanguage. The only necessary background for comprehensive reading of these notes are firstyear calculus and linear algebra. ..."
Abstract
 Add to MetaCart
These are lecture notes for a 1–semester undergraduate course (in computer science, mathematics, physics, engineering, chemistry or biology) in applied categorical metalanguage. The only necessary background for comprehensive reading of these notes are firstyear calculus and linear algebra.