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Based on 12.0 fb−1 of e+e− collision data samples collected by the BESIII detector at center-of-mass energies from 4.1271 to 4.3583 GeV, a partial wave analysis is performed for the process e+e− → π+π−J/ψ. The cross sections for the subprocesses e+e− → π+Zc(3900)− + c.c. → π+π−J/ψ, f0 (980)(→ π+π-)J/ψ, and (π+π-)S-waveJ/ψ are measured for the first time. The mass and width of the Zc(3900)± are determined to be 3884.6 ± 0.7 ± 3.3 MeV/c2 and 37.2 ± 1.3 ± 6.6 MeV, respectively. The first errors are statistical and the second systematic. The final state (π+π-)S-waveJ/ψ dominates the process e+e- → π+π-J/ψ. By analyzing the cross sections of π±Zc(3900)∓ and f0(980)J/ψ, Y(4220) has been observed. Its mass and width are determined to be 4225.7 ± 4.1 ± 3.4 MeV/c2 and 57.5 ± 9.4 ± 12.1 MeV, respectively.