# Extendability of Continuous Maps Is Undecidable

@article{adek2014ExtendabilityOC, title={Extendability of Continuous Maps Is Undecidable}, author={Martin {\vC}adek and Marek Krc{\'a}l and Jir{\'i} Matousek and Luk{\'a}s Vokr{\'i}nek and Uli Wagner}, journal={Discrete \& Computational Geometry}, year={2014}, volume={51}, pages={24-66} }

We consider two basic problems of algebraic topology: the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity.The extension problem is the following: Given topological spaces X and Y, a subspace A⊆X, and a (continuous) map f:A→Y, decide whether f can be extended to a continuous map $\bar{f}\colon X\to Y$. All spaces are given as finite simplicial complexes, and the map f is simplicial.Recent positive algorithmic…

## 29 Citations

### Extending continuous maps: polynomiality and undecidability

- Mathematics, Computer ScienceSTOC '13
- 2013

An algorithm is obtained that solves the extension problem in polynomial time assuming Y (k-1)-connected and dim X ≤ 2k-2, and the algorithm also provides a classification of all extensions up to homotopy (continuous deformation).

### Extending continuous maps : polynomiality and undecidability ( a survey ) ∗

- Mathematics
- 2013

We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of computational complexity. The extension problem asks,…

### Algorithmic Solvability of the Lifting-Extension Problem

- MathematicsDiscret. Comput. Geom.
- 2017

An algorithm is presented that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions, which is new even in the non-equivariant situation.

### Computing All Maps into a Sphere

- MathematicsJ. ACM
- 2014

A computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of such maps, in the stable range.

### Polynomial-Time Homology for Simplicial Eilenberg–MacLane Spaces

- MathematicsFound. Comput. Math.
- 2013

It is shown that the Eilenberg–MacLane space, represented as a simplicial group, can be equipped with polynomial-time homology, and a suitable discrete vector field is constructed, in the sense of Forman’s discrete Morse theory, on $K(\mathbb{Z},1)$.

### Decidability of the Extension Problem for Maps into Odd-Dimensional Spheres

- MathematicsDiscret. Comput. Geom.
- 2017

It is proved that the problem of the existence of a continuous map X→Y extending a given map A→Y, defined on a subspace A⊆X, is undecidable, even for Y an even-dimensional sphere.

### Computing homotopy classes for diagrams

- Mathematics, Computer ScienceArXiv
- 2021

An algorithm is presented that, given finite simplicial sets X, A, Y with an action of a finite group G, computes the set [X,Y ] G of homotopy classes of equivariant maps l : X → Y extending a givenEquivariant map f : A → Y under the stability assumption, which is applicable to the r-Tverberg problem.

### Polynomial-Time Computation of Homotopy Groups and Postnikov Systems in Fixed Dimension

- MathematicsSIAM J. Comput.
- 2014

A polynomial-time solution of the extension problem, where the input consists of finite simplicial complexes X, Y, plus a subspace A and a (simplicial) map f:A to Y, and the question is the extendability of f to all of X.

### Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

- Mathematics
- 2021

We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in R is complete for the Existential Theory of the Reals for all d ≥ 3 and k…

### Robust Satisfiability of Systems of Equations

- Mathematics, Computer ScienceSODA
- 2014

The problem of robust satisfiability of systems of nonlinear equations is studied, and it is proved that the problem is undecidable when dim K ≥ 2n−2, where the threshold comes from the stable range in homotopy theory.

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