The capacity of wedge sum of spheres of different dimensions

Abstract

K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum and raised some interesting questions about it. In this paper, during computing the capacity of wedge sum of finitely many spheres of different dimensions and the complex projective plane, we give a negative answer to a question of Borsuk whether the capacity of a compactum determined by its homology properties.

Connected sums of sphere products and minimally non-Golod complexes

“…is not an isomorphism, and hence the retract R L−{i} does not contain the top cell of R L . Since R L−{i} is then a homotopy retract of ℓ k=1 (S n k ×S n−n k ), we conclude that R L−{i} is homotopy equivalent to a wedge of spheres by [17,Theorem 3.3].…”

“…For this reason, we will need a stronger version of Lemma 3.1. A proof that the statement of Lemma 3.1 still holds without the simply-connectedness hypothesis, provided that the index set I is finite, is given in [17,Theorem 3.3].…”

The capacity of some classes of polyhedra

“…They showed that the capacities (and also the depths) of a compact orientable surface of genus g 0 and a compact non-orientable surface of genus g > 0 are equal to g + 2 and [ g 2 ] + 2, respectively. The authors, in [25] computed the capacities of finite wedges of spheres of various dimensions. Indeed, we showed that the capacity (and also the depth) of n∈I (∨ in S n ) is equal to n∈I (i n + 1), where ∨ in S n denotes the wedge of i n copies of S n , I is a finite subset of N and i n ∈ N. Also, in [24], we computed the capacity of the product of two spheres of the same or different dimensions and the capacities of lens spaces which are a class of closed orientable 3-manifolds.…”

“…Also, we computed the capacity of the wedge of finitely many Moore spaces of different degrees and the capacity of the product of finitely many Eilenberg-MacLane spaces of different homotopy types. In [18], we showed that the capacity of…”

“…They showed that the capacities of a compact orientable surface of genus g ≥ 0 and a compact nonorientable surface of genus g > 0 are equal to g + 2 and [ g 2 ] + 2, respectively. In [18], we proved the capacity of a 2-dimensional CW-complex P with free fundamental group π 1 (P ) is finite and is equal to (rank π 1 (P ) + 1) × (rank H 2 (P ) + 1).…”