Consider the following partial order. The objects are unordered tuples $\{V\_1,\ldots,V\_m\}$, where each $V\_i \subseteq \mathbf{R}^n$ is a nontrivial linear subspace and $V\_1 \oplus \cdots \oplus V\_m = \mathbf{R}^n$. We say that $\{V\_1,\ldots,V\_m\} \geq \{W\_1,\ldots,W\_i\}$ if for each $j=1,\ldots,m$ there exists $1 \leq k \leq i$ such that $V\_j \subseteq W\_k$. (I.e. if $\{V\_1,\ldots,V\_m\}$ is a finer decomposition than $\{W\_1,\ldots,W\_i\}$.) This partial order has a minimal element, $\{\mathbf{R}^n\}$, which is the trivial decomposition. If we remove it, what is the homotopy type of this partial order?


Some things that are known about similar problems.


1. If we're working over $\mathbb{F}\_1$, i.e. with subsets of $\{1,\ldots,n\}$, then there is both a minimal and a maximal object. If we remove both of these then the homotopy type of the partial order is a wedge of $S^{n-3}$'s.
2. If instead we work with ordered decompositions, then this is a barycentric subdivision of Ruth Charney's split building, so it is a wedge of $S^{n-2}$'s.


Edit: clarity and notation.



What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$?
Let me write $V$ for a finite-dimensional vector space over some field (the field will not play a role), and $\mathsf{P}(V)$ for the poset described in the question, which I consider as a category. Let me rephrase the order relation: $\{V\_1, \ldots, V\_n\} \geq \{W\_1, \ldots, W\_m\}$ if and only if each $W\_j$ is a direct sum of $V\_i$'s. The nerve of this poset is $(\mathrm{dim}(V)-2)$-dimensional, so to see that it is a wedge of spheres one must show that it is $(\mathrm{dim}(V)-3)$-connected. I think this is true.


Let $\mathsf{S}(V)$ denote the following category. It has objects given by pairs $([n], f : [n] \to Sub(V))$ consisting of a standard set $[n] := \{1,2,\ldots, n\}$ and a function from $[n]$ to the set of proper vector subspaces of $V$, such that $\bigoplus\_{i \in [n]} f(i) = V$. A morphism in $\mathsf{S}(V)$ from $([n], f)$ to $([n'], f')$ is given by a surjection $e : [n] \to [n']$ such that $f'(i) = \bigoplus\_{j \in e^{-1}(i)} f(j)$.


There is a functor $F : \mathsf{S}(V) \to \mathsf{P}(V)$ given by sending $([n], f)$ to the unordered collection $\{f(i)\}\_{i \in [n]}$.


Now let me do something a bit odd: choose a total order $\prec$ on the set of vector subspaces of $V$. Then we can consider objects of $\mathsf{P}(V)$ as given by lists of subspaces $(V\_1, \ldots, V\_n)$ with $V\_1 \prec V\_2 \prec \cdots \prec V\_n$, and such that $\bigoplus\_{i=1}^n V\_i = V$. Attempt to define a functor $G : \mathsf{P}(V) \to \mathsf{S}(V)$ by
$$G(V\_1, \ldots, V\_n) := ([n], i \mapsto V\_i)$$
on objects. If $(V\_1, \ldots, V\_n) \geq (W\_1, \ldots, W\_m)$ in $\mathsf{P}(V)$ then each $W\_j$ is a direct sum of $V\_i$'s. Define a function $e : [n] \to [m]$ by $e(i) = j$ if $V\_i \subset W\_j$; this gives a morphism in $\mathsf{S}(V)$ from $([n], i \mapsto V\_i)$ to $([n], j \mapsto W\_j)$. To check that this is indeed a functor suppose that $(W\_1, \ldots, W\_m) \geq (U\_1, \ldots, U\_\ell)$, with $e'(j) = k$ when $W\_j \subset U\_k$. Then we indeed have $e' e(i) = k$ when $V\_i \subset W\_{e(i)} \subset U\_k$, so $G$ is indeeed a functor.


The conclusion of this discussion is that $\mathsf{P}(V)$ is a retract of $\mathsf{S}(V)$, so it suffices to show that $\mathsf{S}(V)$ is $(\mathrm{dim}(V)-3)$-connected.


The full name of $\mathsf{S}(V)$ is $\mathsf{S}^{E\_\infty}(V)$, the $E\_\infty$-splitting category defined by Galatius, Kupers, and I in Definition 17.17 of [Cellular $E\_k$-algebras](https://arxiv.org/abs/1805.07184) (applied to the symmetric monoidal groupoid of finite-dimensional vector spaces; the way I have described it here is not identical to that definition, but is an equivalent category). It has a cousin $\mathsf{S}^{E\_1}(V)$ which turns out to be precisely (the category of simplices of) Charney's split building. In particular $\mathsf{S}^{E\_1}(V)$ is $(\mathrm{dim}(V)-3)$-connected by Charney's theorem.


It would take too long to explain all the details here, but the theory developed in that paper shows that the collection of all $\Sigma^2 \mathsf{S}^{E\_\infty}(V)$'s can be obtained from the collection of all $\Sigma^2 \mathsf{S}^{E\_1}(V)$'s by an iterated bar construction, and in particular given Charney's connectivity result (for **all** $V$) it follows that the $\Sigma^2 \mathsf{S}^{E\_\infty}(V)$ are also $(\mathrm{dim}(V)-1)$-connected, so the $\mathsf{S}^{E\_\infty}(V)$ are homologically $(\mathrm{dim}(V)-3)$-connected. (It should not be hard to show it is simply-connected, by hand.) I'm happy to discuss the details by e-mail, if you like.


(Also, now that I believe it is true I expect there must be a more elementary way to deduce it from Charney's theorem.)


Edit:


1. As @inna says in the comments, the functor $F : \mathsf{S}(V) \to \mathsf{P}(V)$ is actually an equivalence, because there is a natural isomorphism $\mathrm{Id} \Rightarrow F \circ G$ given by applying the unique permutation necessary to put things in the order given by $\prec$.
2. Let me try to give some references for what I said in the final paragraph. All references are to [Cellular $E\_k$-algebras](https://arxiv.org/abs/1805.07184):


Writing $\mathsf{G}$ for the symmetric monoidal category of finite-dimensional vector spaces and linear isomorphisms, we work in the category $\mathsf{sSet}\_\*^\mathsf{G}$ of functors from $\mathsf{G}$ to pointed simplicial sets. This is again symmetric monoidal by Day convolution, and the functor
$$\mathbb{t}(V) = \begin{cases}
S^0 & V \neq 0\\
\* & V=0
\end{cases}$$
is a nonnital commutative monoid, and hence also a nonunital $E\_\infty$-algebra. (In the paper this object is called $\underline{\*}\_{>0}$.)


There is another character involved: the derived $E\_k$-indecomposables $Q^{E\_k}\_\mathbb{L}(\mathbb{t})$ for $1 \leq k \leq \infty$, which are again objects of $\mathsf{sSet}\_\*^\mathsf{G}$. For $V$ an object of $\mathsf{G}$ we write $H\_{V,d}^{E\_k}(\mathbb{t}) := H\_d(Q^{E\_k}\_\mathbb{L}(\mathbb{t})(V))$.


The ingredients I have in mind are now:


a. Combining Proposition 17.4 and Lemma 17.10 shows that
$$\Sigma Q^{E\_1}\_\mathbb{L}(\mathbb{t})(V) \simeq \Sigma^2 \mathsf{S}^{E\_1}(V)$$
(in fact by Section 17.5 this can be desuspended once) and hence Charney's theorem shows that $H\_{V,d}^{E\_1}(\mathbb{t})=0$ for $d < \dim(V)-1$.


b. Theorem 14.4 (this is the application of the bar construction result, but is packaged so one doesn't explicitly have to think about that) applied with $\rho(V) := \mathrm{dim}(V)$ shows that $H\_{V,d}^{E\_\infty}(\mathbb{t})=0$ for $d < \dim(V)-1$ too.


c. Corollary 17.23 shows that $Q^{E\_\infty}\_\mathbb{L}(\mathbb{t})(V) \simeq \Sigma \mathsf{S}^{E\_\infty}(V)$ so the above translates to $\tilde{H}\_\*(\mathsf{S}^{E\_\infty}(V))=0$ for $\* \leq \mathrm{dim}(V)-3$.


