Contractibility of a crossing-matching complex

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For a reducible quasi-reduced free-group word ww, let FwF_w be the CW complex whose cells encode crossing matchings and their resolutions. Must FwF_w be contractible?

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No one appears to have made public progress on this contractibility question.

No public discussion or published progress was found for the question of whether the crossing-matching complex associated with a reducible quasi-reduced free-group word is contractible.

Current status (as of March 2026): The question appears open, with no recorded public activity found.

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