# Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems)

## TL;DR

This paper investigates the algebraic conditions determining the complexity of infinite-domain CSPs, showing that certain local polymorphism identities do not guarantee tractability, thus resolving key open problems in the field.

## Contribution

It demonstrates that local satisfaction of height 1 identities by polymorphisms does not imply global satisfaction, providing a negative answer to a major conjecture in infinite-domain CSP complexity classification.

## Key findings

- Counterexample structures with tractable CSPs lacking certain polymorphism identities
- Difference between local and global satisfaction of identities in $\, ext{ω}$-categorical structures
- Resolution of an open problem regarding orbit growth and identity satisfaction

## Abstract

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable when the model-complete core of the template has a pseudo-Siggers polymorphism, and NP-complete otherwise. One of the important questions related to this conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each non-trivial set of height 1 identities a structure whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of non-trivial height 1 identities by polymorphisms of the structure. We show that local satisfaction and global satisfaction of non-trivial height 1 identities differ for $\omega$-categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.