For every simplicial sphere K of dimension n-1 and every characteristic map lambda:V(K)→F_2^n such that, for each facet sigma=v_1,…,v_n of K, the vectors lambda(v_1),…,lambda(v_n) form a basis of F_2^n, does there exist a map widetildelambda:V(K)→Z^n such that widetildelambda(v)bmod 2=lambda(v) for every vertex v and, for every facet sigma=v_1,…,v_n, the vectors widetildelambda(v_1),…,widetildelambda(v_n) form a basis of Z^n (equivalently, their determinant is pm1)?

Toric lifting problem

For every simplicial sphere $K$ of dimension $n-1$ and every characteristic map $\lambda:V(K)\to\mathbb{F}_2^n$ such that, for each facet $\sigma=\{v_1,\ldots,v_n\}$ of $K$ , the vectors $\lambda(v_1),\ldots,\lambda(v_n)$ form a basis of $\mathbb{F}_2^n$ , does there exist a map $\widetilde{\lambda}:V(K)\to\mathbb{Z}^n$ such that $\widetilde{\lambda}(v)\bmod 2=\lambda(v)$ for every vertex $v$ and, for every facet $\sigma=\{v_1,\ldots,v_n\}$ , the vectors $\widetilde{\lambda}(v_1),\ldots,\widetilde{\lambda}(v_n)$ form a basis of $\mathbb{Z}^n$ (equivalently, their determinant is $\pm1$ )?

References

Primary source

arXiv

Additional references

A Finite-Geometric Obstruction and a Polytopal Separation of Buchstaber Invariants — arXiv — Suyoung Choi, Hyeontae Jang

combinatorial-topology toric-topology

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

A new unrefereed preprint claims a counterexample, so the question may have a negative answer, but that claim has not been independently checked.

The problem asks whether every characteristic map over $\mathbb{F}_2$ on a simplicial sphere lifts to an integral characteristic map over $\mathbb{Z}$ . Zhi Lü proposed it at the 2011 Toric Topology conference.

Known results

Every simplicial $2$ -sphere has the toric lifting property (reported in the April 25, 2024 source).
Every $(n-1)$ -dimensional PL sphere with at most $n+4$ vertices has the property (April 25, 2024 source).
Every simplicial $3$ -sphere with at most $2020$ vertices has the property; an earlier bound was $88$ .
Ayzenberg and Sun gave non-liftability results for the universal complex $X(\mathbb{Z}_2^4)$ ; maps with at most $13$ distinct vertex images lift, while sets of size at least $14$ need not.

October 2026 claimed counterexample

On October 6, 2026, Suyoung Choi and Hyeontae Jang’s preprint A Finite-Geometric Obstruction and a Polytopal Separation of Buchstaber Invariants claimed a polytopal sphere whose real and integral Buchstaber numbers differ, giving a negative answer. The claim is unverified.

Current status (as of October 2026): The preprint claims to settle the problem negatively, but without independent verification the general problem remains open in the verified record.

Sources

No solutions have been posted yet.