arXiv subjects

Martin Arkowitz

Publications and source records attributed to Martin Arkowitz.

6 recordsLinked to original sources

arXiv2016

Binary Operations for Homotopy Groups with Coefficients

We define and study binary operations for homotopy groups with coefficients. We give conditions to prove that certain binary operations are the homomorphic image of the generalized Whitehead product. This allows carrying over properties of the generalized Whitehead product to these operations. We discuss two classes of binary operations, the Whitehead products and the Torsion products. We introduce a new class of operations called Ext operations and determine some of its properties. We compare the Torsion product to the Whitehead product in a special case. We prove that the smash product of two Moore spaces has the homotopy type of a wedge of two Moore spaces.

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arXiv2005

Homotopy Actions, Cyclic Maps and their Duals

An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_A: A to X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an H-space that are compatible with the H-structure. As a corollary, we prove that if any two actions F and F' of A on X have cyclic maps f and f' with Omega(f) = Omega(f'), then Omega(F) and Omega(F') give the same action of Omega(A) on Omega(X). We introduce a new notion of the category of a map g and prove that g is cocyclic if and only if the category is less than or equal to 1. From this we conclude that if g is cocyclic, then the Berstein-Ganea category of g is <= 1. We also briefly discuss the relationship between a map being cyclic and its cocategory being <= 1.

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arXiv2004

The cone length and category of maps: pushouts, products and fibrations

For any collection of spaces A, we investigate two non-negative integer homotopy invariants of maps: l_A(f), the A-cone length of f, and L_A(f), the A-category of f. When A is the collection of all spaces, these are the cone length and category of f, respectively, both of which have been studied previously. The following results have been obtained: (1) For a map of one homotopy pushout diagram into another, we derive an upper bound for I_A and L_A of the induced map of homotopy pushouts in terms of I_A and L_A of the other maps. This has many applications including an inequality for I_A and L_A of the maps in a mapping of one mapping cone sequence into another. (2) We establish an upper bound for I_A and L_A of the product of two maps in terms of I_A and L_A of the given maps and the A-cone length of their domains. (3) We study our invariants in a pullback square and obtain as a consequence an upper bound for the A-cone length and A-category of the total space of a fibration in terms of the A-cone length and A-category of the base and fiber. We conclude with several remarks, examples and open questions.

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arXiv2004

The Lefschetz-Hopf theorem and axioms for the Lefschetz number

The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X -->Y and g:Y -->X; (2) if (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r is the projection onto the rth summand. If f:X -->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X -->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X -->X is a selfmap of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f.

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arXiv2004

The sectional category of a map

We study a generalization of the Svarc genus of a fiber map. For an arbitrary collection E of spaces and a map f:X-->Y, we define a numerical invariant, the E-sectional category of f, in terms of open covers of Y. We obtain several basic features of E-sectional category, including those dealing with homotopy domination and homotopy pushouts. We then give three simple properties which characterize the E-sectional category. In the final section we obtain inequalities for the E-sectional category of a composition and inequalities relating the E-sectional category to the Fadell-Husseini category of a map and the Clapp-Puppe category of a map.

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arXiv2001

Homotopy classes that are trivial mod F

If F is a collection of topological spaces, then a homotopy class \alpha in [X,Y] is called F-trivial if \alpha_* = 0: [A,X] --> [A,Y] for all A in F. In this paper we study the collection Z_F(X,Y) of all F-trivial homotopy classes in [X,Y] when F = S, the collection of spheres, F = M, the collection of Moore spaces, and F = \Sigma, the collection of suspensions. Clearly Z_\Sigma (X,Y) \subset Z_M(X,Y) \subset Z_S(X,Y), and we find examples of finite complexes X and Y for which these inclusions are strict. We are also interested in Z_F(X) = Z_F(X,X), which under composition has the structure of a semigroup with zero. We show that if X is a finite dimensional complex and F = S, M or \Sigma, then the semigroup Z_F(X) is nilpotent. More precisely, the nilpotency of Z_F(X) is bounded above by the F-killing length of X, a new numerical invariant which equals the number of steps it takes to make X contractible by successively attaching cones on wedges of spaces in F, and this in turn is bounded above by the F-cone length of X. We then calculate or estimate the nilpotency of Z_F(X) when F = S, M or \Sigma for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as SU(n) and Sp(n). The paper concludes with several open problems.

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