Let L be a loop in the Hawaiian earring and let S be the shift map. If S(L) is homologous to L, then we can construct an element of H_2 by sort of making a cylinder of L on one side and S(L) on the other, and join them at the edge because they're homologous, if that makes sense
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So now we need an L. If a_n is the loop around the nth circle, I think a_1+a_2−a_1+a_3−a_2+a_4−a_3+… works, but I'm having a hard time proving that it's not nullhomologous.
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