$\DeclareMathOperator\Idl{Idl}$Let $P$ be a finite, connected poset with at least two elements, and let $\Idl\_{\neq \emptyset, P}(P)$ be the set of downward closed sets $S \subset P$ such that $S \neq \emptyset$ and $S \neq P$, ordered under inclusion.


**Question:** Is $\Idl\_{\neq \emptyset, P}(P)$ weakly contractible?


**Notes:**


* If $P$ has less than two elements, then $\Idl\_{\neq \emptyset, P}(P)$ is empty. If $P$ is a discrete poset with $\geq 2$ elements, then $|\Idl\_{\neq \emptyset, P}(P)| \simeq S^{|P|-2}$. Hence the restriction to connected posets with $\geq 2$ elements.
* If $P$ has four elements $a,b < c,d$, so that $|P| \simeq S^1$, then $\Idl\_{\neq \emptyset, P}(P)$ has 5 elements arranged in an "$X$" shape, and is weakly contractible.
* In general, if we add $\emptyset$ and $P$ back in, the poset $\Idl(P)$ of all ideals is always contractible (for at least 4 reasons -- it has an initial *and* a terminal object, *and* binary meets *and* binary joins). Similarly, $\Idl\_{\neq \emptyset}(P)$ and $\Idl\_{\neq P}(P)$ are contractible if $P \neq \emptyset$.
* I checked a few random examples with Sage, which at least turned out to have vanishing reduced homology. (Although I'm not sure how much I trust my code!)



Is the poset $\mathrm{Idl}_{\neq \emptyset, P}(P)$ of nonempty, proper ideals in a finite connected poset $P$ (empty or) weakly contractible?
Let me be a little more historical. As mentioned, it is well known that (open intervals of) distributive lattices are shellable. This is mentioned in Corollary 3.2 of Björner's classic paper "Shellable and Cohen-Macaulay partially ordered sets" (<https://doi.org/10.1090/S0002-9947-1980-0570784-2>) but is an older result I believe.


The homotopy type of a shellable complex is a wedge of spheres (of the same dimension). How many spheres do we get? By a basic homology computation, we get $|\mu(\hat{0},\hat{1})|$ of them, where $\mu$ is the Möbius function and $\hat{0}$/$\hat{1}$ are the minimal/maximal elements of the lattice. In a distributive lattice $J(P)$ of order ideals, the Möbius function is given by
$$\mu(I,I') = \begin{cases} (-1)^{\# I' \setminus I} &\textrm{if $I' \setminus I$ is an antichain}, \\ 0 &\textrm{otherwise}. \end{cases}$$
See e.g. Example 3.9.6 of Stanley's EC1, 2nd ed. So your restriction to connected $P$ with $\geq 2$ elements gives $\mu(\hat{0},\hat{1})=0$, hence the open interval $(\hat{0},\hat{1})$ in $J(P)$ is a (homotopy) ball.


