# Not sure whether I find a counterexample to poset fiber theorem

I am trying to implement Theorem 1.1 in the paper "Poset Fiber Theorems" by Bjorner, Wachs, and Welker. https://www.researchgate.net/publication/228782786_Poset_fiber_theorems

I am pretty excited to learn about the theorem, which is a generalization of the well-known Quillen fiber lemma and should be very useful for computation. But when I implement the theorem to the following example, the result seems inconsistent with my knowledge:

The posets are as shown in their Hasse diagrams and how the map is defined is: fixing $$3,6,9,10$$, mapping $$11$$ to a new minimum point $$0$$, and mapping other points downward by 1 level to the unique choice in $$\{3,6,9,10\}$$. From the construction, it is order-preserving and hence a poset map.

The conditions of Theorem 1.1 are satisfied as shown in the table on the right. Since $$Q$$ is the cone over the order complex $$\Delta(\{3,6,9,10\})$$, it is contractible. Thus, applying Theorem 1.1 in the paper, $$\Delta(P)$$, the order complex of $$P$$, is homotopic equivalent to a wedge of $$\Delta(f^{-1}(Q_{\leq q}))\star\Delta(Q_{>q})$$, where $$\star$$ denotes the join operation.

I am only interested in the homology group over fields, so I will use Betti numbers for the rest of the discussion. I also use Kunneth's formula for the join of simplicial complexes (see, e.g., the very beginning of https://arxiv.org/abs/math/0412552) in the case of field coefficients:

For $$q = 0$$, the reduced Betti numbers for $$\Delta(f^{-1}(Q_{\leq q}))\star\Delta(Q_{>q})$$ are all zeros.

For $$q = 9$$ or $$10$$, the reduced Betti numbers are $$0,0,1$$ (i.e. $$H_k$$ for $$k = 0, 1, 2$$).

For $$q = 3$$ or $$6$$, $$f^{-1}(Q_{\leq q})$$ is isomorphic to the face poset of a $$2$$-simplex. Hence, the reduced Betti numbers of $$\Delta(f^{-1}(Q_{\leq q}))\star\Delta(Q_{>q})$$ are all zeros.

Thus, by Theorem 1.1 in the paper, the reduced Betti numbers of $$\Delta(P)$$ are 0, 0, 2 (i.e. $$H_k$$ for $$k = 0, 1, 2$$) and $$0$$ for other dimensions.

However, it can be noted that $$P$$ is isomorphic to the face poset of the CW complex obtained by taking two copies of a $$2$$-simplex and identifying only their corresponding vertices. (You may visualize it as a pair of panties.) Thus, the reduced Betti numbers are 0, 2, 0 (i.e. $$H_k$$ for $$k = 0, 1, 2$$) and $$0$$ for other dimensions. I also verify the result by computer.

I have been checking logical bugs the whole day and cannot find a mistake. I am relatively new to these poset topology things and afraid that I have made some naive/stupid mistakes. Any comments, questions, suspicion about my computations, or pointing out my mistakes are welcome. Thank you.

• The preimage of $Q_{\le 3}$ should contain everything NOT mapped to $6$. Aren't you missing $7$ and $8$? (same for $6$ instead of $3$) – j.p. Jul 29 '20 at 9:01
• I think you are right. LOL Thanks for pointing out. This saves me a lot of time. – Min Wu Jul 29 '20 at 13:45