Suspension of the Lusternik-Schnirelmann Category

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.

Products in homotopy groups with certain prime coefficients

1968 ◽  
Vol 64 (1) ◽  
pp. 11-14
Author(s):  
Paul Green
Keyword(s):  
Homotopy Type ◽  
Group Structure ◽  
Homotopy Groups ◽  
Homotopy Classes ◽  
Image Position ◽  
Simply Connected

Let p be a prime and n ≥ 3. Then there is a simply connected CW-complex, , unique up to homotopy type, such thatLet X be a CW-complex. Write , the group of pointed homotopy classes of pointed maps from to X. The group structure derives from the fact that, under the restriction on n, is a suspension.

Maps which Induce the Zero Map on Homotopy

1968 ◽  
Vol 20 ◽  
pp. 759-768
Author(s):  
C. S. Hoo
Keyword(s):  
Homotopy Type ◽  
Homotopy Classes ◽  
Base Points ◽  
Cw Complexes ◽  
Simply Connected

In this paper, all spaces will have the homotopy type of simply connected CW-complexes, and will have base points which are preserved by maps and homotopies. We denote by [X, Y] the set of homotopy classes of maps from X to Y, and by N[X, Y] the subset of those homotopy classes [ƒ] which induce the zero homomorphism on homotopy, that is, is the zero homomorphism for each i.

Isotopy invariants of topological spaces

1960 ◽  
Vol 255 (1282) ◽  
pp. 331-366 ◽  
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.

Some properties of Hopf-type constructions

1995 ◽  
Vol 117 (2) ◽  
pp. 287-301 ◽  
Author(s):  
Martin Arkowitz ◽  
Paul Silberbush
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Smash Product ◽  
Homotopy Classes ◽  
One To One

If f: X × Y → Z is a map, then the classical Hopf construction associates to f a map hf: X * Y → ΣZ, where X * Y is the join of X and Y and ΣZ the suspension of Z. Since X * Y has the homotopy type of Σ(X Λ Y), the suspension of the smash product of X and Y, the homotopy class of hf can be regarded as an element Hf ↦ [Σ(X Λ Y), ΣZ]. Now elements of [Σ(X Λ Y), ] are in one to one correspondence with homotopy classes in the group [σ(X Λ Y), ΣZ] which are trivial on the suspension of the wedge Σ(X ≷ Y).

An Inequality Between Numerical Homotopy Invariants

1968 ◽  
Vol 20 ◽  
pp. 1295-1299
Author(s):  
M. J. M. Priddis
Keyword(s):  
Topological Space ◽  
Nilpotency Class ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Homotopy Invariants

In (1), Berstein and Ganea denned the nilpotency class of a based topological space. For a based topological space X we write nil X for the nilpotency class of the group ΩX in the category of based topological spaces and based homotopy classes. Hilton, in (3), defined the nilpotency class, nil class K of a based semi-simplicial (s.-s.) complex; actually, the restriction of connectedness can be removed. Hence, by using the total singular complex functor S, an invariant (nil class SX) can be defined for a based topological space X.

Homotopical Nilpotency of Loop-Spaces

1969 ◽  
Vol 21 ◽  
pp. 479-484 ◽  
Author(s):  
C S. Hoo
Keyword(s):  
Homotopy Class ◽  
Base Point ◽  
Loop Spaces ◽  
Homotopy Classes ◽  
Base Points ◽  
Cw Complexes

In this paper we shall work in the category of countable CW-complexes with base point and base point preserving maps. All homotopies shall also respect base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class. Given spaces X, Y, we denote the set of homotopy classes of maps from X to Y by [X, Y]. We have an isomorphism τ: [∑X, Y] → [X, Ω Y] taking each map to its adjoint, where ∑ is the suspension functor and Ω is the loop functor. We shall denote τ(1 ∑x) by e′ and τ-1(1Ωx) by e.

Homotopy of Natural Transformations

1970 ◽  
Vol 22 (2) ◽  
pp. 332-341 ◽  
Author(s):  
K. A. Hardie
Keyword(s):  
Full Subcategory ◽  
Natural Transformation ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Base Points ◽  
Natural Transformations ◽  
Image Position

Let C be a full subcategory of T, the category of based topological spaces and based maps, and let Cn be the corresponding category of n-tuples. Let S, T: Tn → T be covariant functors which respect homotopy classes and let u, v: S → T be natural transformations, u and v are homotopic inC, denoted u ≃ v(C), if uX ≃ vX: SX → TX (X ∈ Cn), that is to say, for every X ∈ C, uX and vX are homotopic (all homotopies are required to respect base points), u and v are naturally homotopic inC, denoted u ≃n v; (C), if there exist morphismssuch that, for every X ∈ C, utX is a homotopy from uX to vX and such that, for every t ∈ I, ut:S → T is a natural transformation.

Non-singular bilinear maps and stable homotopy classes of spheres

1977 ◽  
Vol 82 (3) ◽  
pp. 419-425 ◽  
Author(s):  
Kee Yuen Lam
Keyword(s):  
Homotopy Class ◽  
Stable Homotopy ◽  
Background Information ◽  
Bilinear Maps ◽  
Bilinear Map ◽  
Homotopy Classes ◽  
Image Position

A bilinear map ø Ra x Rb → Rc is non-singular if ø (x, y) = 0 implies x = 0 or y = 0. For background information on such maps see (4, 5, 6, 14). If we apply the ‘Hopf construction’ to ø, we get a mapdefined by 2ø(x, y)) for all x ∈ Ra, y ∈ Rb satisfying ∥x∥2 + ∥y∥2 = 1. Homotopically, Jø coincides with the map obtained by first restricting and normalizing ø to , and then applying the standard Hopf construction ((13), p. 112). In any case, one gets an element [Jø] in , which in turn determines a stable homotopy class of spheres {Jø} in the d-stem , where d = a + b − c −1. An element in which equals {Jø} for some non-singular bilinear map ø will be called bilinearly representable. The first purpose of this paper is to prove

On the 1-dimensional category of Cartesian products

1960 ◽  
Vol 56 (4) ◽  
pp. 425-426
Author(s):  
I. Berstein
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Closed Path ◽  
Cartesian Products ◽  
Unsuccessful Attempt ◽  
Lusternik Schnirelmann ◽  
Image Position ◽  
Simply Connected

The following example arose out of an unsuccessful attempt to improve a result by Ganea and Hilton (5), according to whichimplies that for each field of coefficients F one at least of the factors Xi is acyclic; here cat X is the Lusternik-Schnirelmann category of X. It was natural to ask whether this assumption also implies that one at least of the Xi is simply connected. This could be proved if it were possible to obtain spaces of arbitrarily large 1-dimensional category ((3), (4)) by forming Cartesian products of sufficiently many 2-dimensional pseudoprojective spaces (1) corresponding to different primes. However, we prove that this is impossible:Theorem. If P1, …, Pn are 2-dimensional pseudo-projective spaces corresponding to different primes p1, …, pn, thenBefore proving this theorem, we recall the definitions involved. The 1-dimensional category cat1X of a space X is the least number of open sets covering X and such that each closed path in any of them is contractible in X. A 2-dimensional pseudo-projective space P corresponding to the prime p is the complex obtained from the disk |z| ≤ 1 by identifying the points ei[θ+(2mπ|p)], m = 0, 1, …, p − 1. We have (1)for k ≥ 2 and cat1P = 3 since π1(P) is not free and cat1P ≤ 1 + dim P (4).

Cyclic Maps From Suspensions to Suspensions

1972 ◽  
Vol 24 (5) ◽  
pp. 789-791 ◽  
Author(s):  
C. S. Hoo
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Homotopy Classes ◽  
Whitehead Product ◽  
Base Points ◽  
Cyclic Map ◽  
Cyclic Maps

In [7] Varadarajan denned the notion of a cyclic map f : A → X. The collection of all homotopy classes of such cyclic maps forms the Gottlieb subset G(A, X) of [A, X]. If A = S1 this reduces to the group G(X, X0) of Gottlieb [5]. We show that a cyclic map f maps ΩA into the centre of ΩX in the sense of Ganea [4]. If A and X are both suspensions, we then show that if f : A → X maps ΩA into the centre of ΩX, then f is cyclic. Thus for maps from suspensions to suspensions, Varadarajan's cyclic maps are just those maps considered by Ganea. We also define G (Σ4, ΣX) in terms of the generalized Whitehead product [1], This gives the computations for G(Sn+k, Sn) in terms of Whitehead products in π2n+k(Sn).We work in the category of spaces with base points and having the homotopy type of countable CW-complexes. All maps and homotopies are with respect to base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class.

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