2 edition of **Invariants for effective homotopy classification and extension of mappings** found in the catalog.

Invariants for effective homotopy classification and extension of mappings

Paul Olum

- 88 Want to read
- 37 Currently reading

Published
**1961**
by American Mathematical Society in Providence
.

Written in English

- Algebraic topology.,
- Invariants.

**Edition Notes**

Other titles | Homotopy classification and extension of mappings. |

Statement | by Paul Olum. |

Series | Memoirs of the American Mathematical Society, no. 37, Memoirs of the American Mathematical Society -- no. 37. |

The Physical Object | |
---|---|

Pagination | 69 p. |

Number of Pages | 69 |

ID Numbers | |

Open Library | OL14119171M |

Hans-Joachim Baues's 72 research works with citations and 1, reads, including: The DG-Category of Secondary Cohomology Operations. homotop y invariants w ork in simple (mainly low-dimensional) situations. Homotopy and homotopy equi valence Homotopy of maps. It is interesting to point out that in order to deÞne the homotop y equi valence, a relation between spaces, we Þrst need to consider a.

Recent work has further shown the utility of the classifying space for the homotopy classification of maps, relating the classical group theory of abstract kernels, and their obstructions to extensions, with mappings of an n-dimensional space into a space whose homotopy vanishes between 1 and n. Here the notion of fibration of crossed complexes. Full text Full text is available as a scanned copy of the original print version. Get a printable copy (PDF file) of the complete article (K), or click on a page image below to browse page by page.

IÎ COMBINATORIAL HOMOTOPY 4>: Kn—>X, of every finite simplicial complex, Kn, of at most n dimen sions. We alter this by defining f^ng if, and only if, /map:Kn—>X, of every CW-complex, Kn, of at most n dimensions. As in [ó] we describe a map, f".X—»F, as an w-homotopy equivalence if, and only if, it has an n-homotopy inverse, meaning a map, g: Y—>X. Thus, for connected 1-manifolds, two invariants, compactness and presence of boundary, form a complete system of topological invariants. Each of the invariants takes two values. Theorems and above solve the topological classification problem for 1-manifolds in the most effective way that one can desire. Surprisingly, many Topology.

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Invariants for effective homotopy classification and extension of mappings. Providence, R.I.: American Mathematical Society, (OCoLC) Document Type: Book: All Authors / Contributors: Paul Olum. Genre/Form: Electronic books: Additional Physical Format: Print version: Olum, Paul.

Invariants for effective homotopy classification and extension of mappings. These invariants, together with the Betti numbers and coefficients of torsion, characterize the homotopy type of one of these polyhedra. They are also applied to the classification of continuous mappings of an (n+ 2)-dimensional polyhedron into an (n+ 1)-sphere (n> 2).

The classification largely matches the one recently obtained (arXiv) for a similar setting leaving out translation invariance. However, the translation invariant case has some finer distinctions, because some walks may be connected only by breaking translation invariance along the way, retaining only invariance by an even number of Cited by: Paul Olum, Invariants for effective homotopy classification and extension of mappings, Mem.

Amer. Math. Soc. 37 (), MR [9] Jean-Pierre Serre, Cohomologie modulo 2 des complexes d’Eilenberg-MacLane, Comment. Robert E. Mosher and Martin C. Tangora, Cohomology operations and applications in homotopy theory, Harper & Row, Publishers, New York-London, MR ; Paul Olum, Invariants for effective homotopy classification and extension of mappings, Mem.

Amer. Math. Soc. 37. This chapter discusses with a part of the converse problem of the determination of the homotopy type by algebraic invariants, and shows in effect that the only one-and two-dimensional invariants that enter are the fundamental group π 1, the second homotopy group π 2, and a certain three-dimensional co-homology class of π 1 in π 2.

A functor on spaces (e.g. some cohomology functor) is called “homotopy invariant” if it does not distinguish between a space X and the space X \times I, where I is an interval; equivalently if it takes the same value on morphisms which are related by a (left) homotopy.

The book contains, as well as lattices and reflection groups, the classification of complex analytic surfaces, the Torelli-type theorem, the subjectivity of the period map, Enriques surfaces, an application to the moduli space of plane quartics, finite automorphisms of K3 surfaces, Niemeier lattices and the Mathieu group, the automorphism group.

In higher dimensions, the rational Pontrjagin classes are topological invariants by a fundamental theorem of Novikov. These can be varied more or less arbitrarily within a homotopy type, subject to the condition that the polynomial in the Pontrjagin classes (the L.

Part I. Derived functors and homotopy (co)limits 1 Chapter 1. All concepts are Kan extensions 3 Kan extensions 3 A formula 5 Mapping spaces Chapter Simplicial categories and homotopy coherence The goal of this book is to use category theory to illuminate abstract ho.

Postnikov invariants of a homotopy type 54 Boundary invariants of a homotopy type 63 Homotopy decomposition and homology decomposition 72 Unitary invariants of a homotopy type 76 Chapter 3 On the classification of homotopy types 81 kype functors 81 bype functors 85 Duality of bype and kype 89 The classification.

Homotopy Class Homotopy Group Link Group Homotopy Classification General Topological Space These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

studying a map in an atlas. The basic pattern of classiﬁcation, and at the same time the guarantee for its stability, is typically homotopy theory. That is, we consider two systems to be equivalent, or “in the same topological phase”, if one can be deformed continuously into the other while retaining somekeyproperties.

Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share. If we have a homotopy H: X × [0,1] → Y and a cover p: Y → Y and we are given a map h 0: X → Y such that H 0 = p h 0 (h 0 is called a lift of h 0), then we can lift all H to a map H: X × [0,1] → Y such that p H = H.

The homotopy lifting property is used to characterize fibrations. Another useful property involving homotopy is the homotopy extension property, which characterizes.

Homotopy invariance of -invariants. extensions. includes. all. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the.

B Homotopy invariants Homotopy groups Let M (X, Y) denote the set of continuous mappings between the topological spaces X and Y.

Two mappings f, g ∈ M (X, Y) are called homotopic if there is a one-parameter family of mappings ft ∈ M (X, Y) depending continuously on t ∈ [0, 1] and joining f and g, i.e., such that f0 = f while f1 = g. The author, a leading figure in algebraic topology, provides a modern treatment of a long established set of questions in this important research area.

The book's principal objective--and main result--is the classification theorem on k-variants and boundary invariants, which supplement the classical picture of homology and homotopy groups, along with computations of types that are obtained by.

These homotopy invariants can be considered as obstructions for extensions of covers of a subspace A to a space X. We using these obstructions for generalizations of the classic KKM (Knaster-Kuratowski-Mazurkiewicz) and Sperner lemmas.

In particular, we show that in the case when A is a k-sphere and X is a (k+1)-disk there exist KKM type lemmas. We provide a classification of translation invariant one-dimensional quantum walks with respect to continuous deformations preserving unitarity, locality, translation invariance, a gap condition, and some symmetry of the tenfold way.

The classification largely matches the one recently obtained (arXiv) for a similar setting leaving out translation invariance.

However, the translation. -invariants are generally characterized by Chern numbers or winding numbers defined from the Berry curvature of the system and -invariants correspond to Kane–Mele invariants or Chern–Simons invariants [7, 8]. Classification of topological phases in different dimensions and symmetry classes has been obtained by using Clifford algebras and K.

But the higher homotopy groups π n (S 1) ≃ * \pi_n(S^1) \simeq *, n > 1 n \gt 1 all vanish (and so is a homotopy 1-type). This can be deduced from the result that the loop space Ω S 1 \Omega S^1 of the circle is the group ℤ {\mathbb{Z}} of integers and that S 1 S^1 is path-connected.