Homotopy invariants of foliations

This paper contributes new numerical invariants to the topology of a certain class of polyhedra. 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).

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On some homotopy invariants of pseudoriemannian metrics

Topology - Lecture Notes in Mathematics ◽

10.1007/bfb0099949 ◽

1984 ◽

pp. 341-347

Author(s):

A. S. Miscenko

Keyword(s):

Homotopy Invariants

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Isotopy invariants of topological spaces

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1960.0071 ◽

1960 ◽

Vol 255 (1282) ◽

pp. 331-366 ◽

Cited By ~ 10

Keyword(s):

Homotopy Type ◽

Algebraic Topology ◽

Complete System ◽

Topological Spaces ◽

Recent Developments ◽

The Difference ◽

Linear Graphs ◽

Homotopy Invariants ◽

General Method

A few decades ago, topologists had already emphasized the difference between homotopy and isotopy. However, recent developments in algebraic topology are almost exclusively on the side of homotopy. Since a complete system of homotopy invariants has been obtained by Postnikov, it seems that hereafter we should pay more attention to isotopy invariants and new efforts should be made to attack the classical problems. The purpose of this paper is to introduce and study new algebraic isotopy invariants of spaces. A general method of constructing these invariants is given by means of a class of functors called isotopy functors. Special isotopy functors are constructed in this paper, namely, the m th residual functor R m and the m th. enveloping functor E m . Applications of these isotopy invariants to linear graphs are given in the last two sections. It turns out that these invariants can distinguish various spaces belonging to the same homotopy type.

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Some non-embedding theorems for the Grassmann manifolds G2,n and G3,n

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500026249 ◽

1977 ◽

Vol 20 (3) ◽

pp. 177-185 ◽

Cited By ~ 12

Author(s):

V. Oproiu

Keyword(s):

Projective Spaces ◽

Embedding Problem ◽

Embedding Theorems ◽

Euclidean Spaces ◽

Grassmann Manifolds ◽

Negative Results ◽

Homotopy Invariants

In recent years, the problem of embedding the projective spaces in Euclidean spaces was studied very much, by different methods. Usually, the negative results on the embedding problem are proved by using suitable homotopy invariants. The best known example of such homotopy invariants is given by the Stiefel–Whitney classes.

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Suspension of the Lusternik-Schnirelmann Category

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-131-6 ◽

1970 ◽

Vol 22 (6) ◽

pp. 1129-1132

Author(s):

William J. Gilbert

Keyword(s):

Homotopy Type ◽

Homotopy Class ◽

Category Structure ◽

Topological Spaces ◽

Homotopy Classes ◽

Cw Complexes ◽

Lusternik Schnirelmann ◽

Image Position ◽

Simply Connected ◽

Homotopy Invariants

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],We prove thatandIt is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

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