Homotopy and homology groups of the n-dimensional Hawaiian earring

Abstract

Homotopy and homology groups of the n-dimensional Hawaiian earring by Katsuya E d a (Tokyo) and Kazuhiro K a w a m u r a (Tsukuba) Abstract. For the n-dimensional Hawaiian earring H n , n ≥ 2, π n (H n , o) Z ω and π i (H n , o) is trivial for each 1 ≤ i ≤ n−1. Let CX be the cone over a space X and CX ∨CY be the one-point union with two points of the base spaces X and Y being identified to a point. Then H n (X ∨ Y) H n (X) ⊕ H n (Y) ⊕ H n (CX ∨ CY) for n ≥ 1.

The n-dimensional Spanier group

“…In the following, we state some definitions about the open covering of topological space. (6) An open cover of X is a family {U i : i ∈ I}, of open subsets of X, whose union is the whole set X. (7) An open cover of X by pointed sets is a family {(U i , x i ) : i ∈ I} of pointed subsets, where…”

“…By Theorem 4.14, π uSp n (H n , 0) is contained in the kernel of ϕ : π n (H n , 0) −→ πn (H n , 0). But Eda et al in [6] showed that ϕ is injective, so…”

Inverse limit slender groups

“…Let H n be the one-point union of a sequence of spheres in R n`1 whose radii converge to zero (so that H 1 is just the Hawaiian earring), and let S n denote the unit sphere in R n`1 . Eda and Kawamura [15] showed that for n ą 1 the group π n pH n q is isomorphic to Z N by a homomorphism taking e i P Z N to the homotopy class of embedding of S n onto the i-th sphere of H n .…”

Sequential $n$-connectedness and infinite factorization in higher homotopy groups

“…For instance, Eda's remarkable classification theorem [10] asserts that homotopy types of one-dimensional Peano continua are completely determined by the isomorphism type of their fundamental groups [10]. With the exception of the asphericity of one-dimensional [8] and planar spaces [7] and the work of Eda-Kawamura [12] on shrinking wedges of spaces like the n-dimensional Hawaiian earring H n , essentially no significant progress has been made in the effort to further characterize higher homotopy groups that admit natural infinite product operations. This stagnation is primarily due to the lack of techniques for suitably constructing homotopies on higher dimensional domains.…”

“…In this paper, we overcome the lack of classical homological methods by constructing required homotopies by "brute force." Indeed, Eda and Kawamura's work [12] has also suggested that there is little hope of circumventing such technicalities. These homotopies will typically deform infinitely many parts of a map simultaneously and so the primary difficulty is identifying a construction that is actually continuous.…”