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Apr 11, 2016 at 17:51 comment added Vidit Nanda If there are no zigzag paths of the form $$ c > y_0 < x_0 > y_1 < x_1 > \cdots > y_k < x_k > c' $$ where $c$ and $c'$ are critical $7$ and $5$-dimensional simplices while $(y_\bullet < x_\bullet)$ lie in $\mathcal{M}$, then you definitely have homotopy equivalence with the wedge of spheres. Just be careful that all $>$ signs in the paths indicate "face of", not just "codimension 1 face" as in Forman's description. This will follow either from Liviu's answer or mine.
Apr 11, 2016 at 17:35 answer added Vidit Nanda timeline score: 7
Apr 9, 2016 at 10:33 answer added Liviu Nicolaescu timeline score: 6
Apr 9, 2016 at 9:55 history asked Leo CC BY-SA 3.0