The Singular Homology of the Hawaiian Earring

Cotorsion-free groups from a topological viewpoint

“…As in [16], it suffices to show that H 1 (C(H o ) ∨ C(H e )) is (i) torsion-free; (ii) algebraically compact; (iii) contains a subgroup isomorphic to 2 ℵ 0 Q; and (iv) contains a pure subgroup isomorphic to 2 ℵ 0 Z. The fact that A = H 1 (C(H o ) ∨ C(H e )) is torsion-free follows from the formula …”

Milnor–Thurston homology groups of the Warsaw Circle

“…Singular homology theory is well understood for tame spaces (e.g. simplicial complexes, manifolds) and some steps to grasp its behaviour for more complicated spaces has been taken by a new emerging field, which might be described as "algebraic topology of non-tame spaces" (as it was called in [14] or, more briefly, as "wild algebraic topology", as it was called in [6]; this field is focused on spaces like: Hawaiian Earring [11] [5] [7], Harmonic Archipelago [3] [9], Sierpinski-gasket [1], Griffiths' space [4], etc. ).…”

“…The previously mentioned Hawaiian Earring whose first singular homology group has been algebraically described in [7] is a sequence of planar circles tangential to each other at the single point which have its diameters converging to zero. The reason why this fairly simple space has a very complicated homology group is that chains in singular homology consist of a finite number of simplices, and we clearly see that the Hawaiian Earring has infinite number of "cycles" (and unlike in the case of a one point union of countably many circles with CW-topology there exist paths that wind around infinitely many circles).…”

Low dimensional homotopy groups of suspensions of the Hawaiian earring

“…where H 1 (H 1 ) has been computed in [8]. The space H n is (n − 1)-connected and π n ( H n ) ∼ = H n ( H n ), and again the next nontrivial homotopy group to be computed is π n+1 ( H n ).…”

“…The (n − 1)-fold reduced suspension H n = Σ n−1 o H 1 is a compact metric space whose underlying set is the onepoint union of countably many n-dimensional spheres at the base point o n , and is called the n-dimensional Hawaiian earring. The singular homology of the space H n is complicated ( [1], [8]), and this paper is an attempt to understand the low dimensional homotopy groups of H n . The space H n is (n − 1)-connected and it is shown in [9] that for each n ≥ 2, H n (H n ) ∼ = π n (H n ) ∼ = Z ω , the countable product of the integers.…”