Cohomology ring of tree braid groups and exterior face rings

For a tree T T and positive integer alttext="n"> n encoding="application/x-tex">n , let B Subscript n Baseline upper B encoding="application/x-tex">B_nT denote the -strand braid group on . We use discrete Morse theory techniques to show that cohomology ring H Superscript asterisk left-parenthesis T right-parenthesis"> H ? ( stretchy="false">) encoding="application/x-tex">H^*(B_nT) is encoded by an explicit abstract simplicial complex K K encoding="application/x-tex">K_nT measures -local interactions among essential vertices of that, in many cases (for instance when binary tree), exterior face determined