Hatcher-Algebraic-Topology · Problem 38

Homotopy Theory

Mathematics — Graph Theory · pi3-wedge-not-finitely-generated

Problem

Show π3(S1∨S2)\pi_3(S^1 \vee S^2) is not finitely generated as a module over Z[π1(S1∨S2)]\mathbb{Z}[\pi_1(S^1 \vee S^2)] by considering Whitehead products in the universal cover, using the results in Example 4.52. Generalize this to πi+j−1(S1∨Si∨Sj)\pi_{i+j-1}(S^1 \vee S^i \vee S^j) for i,j>1i, j > 1.

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