Topological representations of matroid maps

NEW

Problem: Reconstructed statement: for a finite noncontractible CW complex $X$, a common integer $\rho$, and $\rho$-immersed matroids $(M,l)$, $(N,l')$, if

$$\mathcal T_X(M,l)\simeq \mathcal T_X(N,l')$$

and there is a surjective weak map $\tau:M\to N$, then the induced map of lattices of flats

$$\tau^\#:\mathcal L(M)\to \mathcal L(N),\qquad F\mapsto \operatorname{cl}_N(\tau(F))$$

is an isomorphism. Here a $\rho$-immersion is a rank- and order-reversing map $\mathcal L(M)\to B_\rho$. The text does not require $\tau$ to be $\rho$-admissible; the counterexample below is admissible anyway.

Result: The conjecture is false.

Let $X$ be the Poincaré homology 3-sphere with an open 3-ball removed. Then $X$ is a finite CW complex, is acyclic, and is not contractible: Mayer-Vietoris gives $\widetilde H_*(X;\mathbb Z)=0$, while van Kampen gives $\pi_1(X)\cong \pi_1$ of the Poincaré sphere, which is nontrivial.

Since $X$ is connected and acyclic, both $\Sigma X$ and $X*X$ are simply connected acyclic CW complexes, hence contractible by Whitehead’s theorem.

Now take $\rho=3$. Let

$$M=U_{2,3}$$

on $E=\{1,2,3\}$, and let $N$ be the rank-2 matroid on $E$ in which $1$ and $2$ are parallel and $3$ is not parallel to them. Thus

$$\mathcal L(M)=\{\varnothing,\{1\},\{2\},\{3\},E\},$$

while

$$\mathcal L(N)=\{\varnothing,\{1,2\},\{3\},E\}.$$

Use the canonical $3$-immersions

$$l(F)=[3-\operatorname{rk}_M(F)],\qquad l'(G)=[3-\operatorname{rk}_N(G)].$$

For a $\rho$-immersion of a rank-2 matroid, Engström’s homotopy formula gives

$$\mathcal T_X(A,l_A)\simeq \bigvee^{w_1(A)} X^{*2}\ \vee\ \bigvee^{w_2(A)} \Sigma X .$$

Both $X^{*2}$ and $\Sigma X$ are contractible, so

$$\mathcal T_X(M,l)\simeq * \simeq \mathcal T_X(N,l').$$

The identity map $\tau:E\to E$ is a surjective weak map $M\to N$, because

$$\operatorname{rk}_N(S)\le \operatorname{rk}_M(S)$$

for every $S\subseteq E$. But

$$\tau^\#(\{1\})=\operatorname{cl}_N(\{1\})=\{1,2\} =\operatorname{cl}_N(\{2\})=\tau^\#(\{2\}),$$

so $\tau^\#$ is not injective and therefore not an isomorphism.

Thus all hypotheses hold, but the conclusion fails. The defect is that “$X$ noncontractible” allows acyclic noncontractible complexes; after one $X$-suspension, the representation can become contractible and lose all matroid information.

Citation: Counterexample appears not to be from the literature. It uses Engström/Stamps’ homotopy formula as stated in Matthew T. Stamps, “Topological representations of matroid maps,” J. Algebraic Combin. 37 (2013), 213–237; arXiv:1104.4152.