Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension (1211.3093v2)

Abstract: For several computational problems in homotopy theory, we obtain algorithms with running time polynomial in the input size. In particular, for every fixed k>1, there is a polynomial-time algorithm that, for a 1-connected topological space X given as a finite simplicial complex, or more generally, as a simplicial set with polynomial-time homology, computes the k-th homotopy group \pi_k(X), as well as the first k stages of a Postnikov system of X. Combined with results of an earlier paper, this yields a polynomial-time computation of [X,Y], i.e., all homotopy classes of continuous mappings X -> Y, under the assumption that Y is (k-1)-connected and dim X < 2k-1. We also obtain a polynomial-time solution of the extension problem, where the input consists of finite simplicial complexes X,Y, where Y is (k-1)-connected and dim X < 2k, plus a subspace A\subseteq X and a (simplicial) map f:A -> Y, and the question is the extendability of f to all of X. The algorithms are based on the notion of a simplicial set with polynomial-time homology, which is an enhancement of the notion of a simplicial set with effective homology developed earlier by Sergeraert and his co-workers. Our polynomial-time algorithms are obtained by showing that simplicial sets with polynomial-time homology are closed under various operations, most notably, Cartesian products, twisted Cartesian products, and classifying space. One of the key components is also polynomial-time homology for the Eilenberg--MacLane space K(Z,1), provided in another paper by Krcal, Matousek, and Sergeraert.

• The paper presents polynomial-time algorithms to compute homotopy groups and construct Postnikov systems for 1-connected spaces, offering a significant efficiency improvement over prior superexponential methods.
• These algorithms provide a polynomial-time solution to the extension problem for maps under dimensional constraints, impacting applications such as robust satisfiability and Z2-index computations.
• The research navigates technical complexities, like managing products of simplicial groups to preserve polynomial-time homology, leveraging tools such as the basic perturbation lemma.

Analyzing Polynomial-Time Computation of Homotopy Groups and Postnikov Systems in Fixed Dimensions

The paper "Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension" presents algorithmic advancements in algebraic topology, focusing on efficient computation of homotopy groups and Postnikov systems for 1-connected topological spaces. These topics are crucial in understanding the homotopy classes of continuous maps between topological spaces, a central subject in algebraic topology.

Analytical Approach to Postnikov Systems

The research elaborates on developing polynomial-time algorithms that compute the $k$th homotopy group $\pi_k(X)$ and construct the first $k$ stages of a Postnikov system for a given 1-connected space $X$. Given a finite simplicial complex or a simplicial set that allows for polynomial-time computation of homology, the algorithms can achieve polynomial-time results for these computations. This is in contrast to earlier algorithms that were considered superexponential and computationally inefficient, such as the foundational methods by Brown.

Implications and Utility

The implications of this research extend beyond homotopy group computation. It provides tools for solving the extension problem, investigating whether a map defined on a subspace can be extended to a larger space. By employing Postnikov systems, the paper demonstrates a method to test extendability of maps in polynomial time, given certain dimensional constraints. The ability to efficiently decide extendability is crucial for applications like the robust satisfiability problem and $Z_2$-index computation, impacting fields such as graph theory (e.g., Lovász's result related to Kneser's conjecture) and manifold embeddability.

Technical Accomplishments and Challenges

The paper embraces technical complexity to ensure the polynomial nature of computations. Utilizing properties of simplicial sets, locally polynomial-time computations, and leveraging both classical topological concepts (e.g., fiber bundles) and sophisticated algebraic frameworks (e.g., chain complexes), it constructs an extensive roadmap for algorithmic advancement. A notable theoretical hurdle in the paper is managing products and combinations of simplicial groups to preserve polynomial-time homology, an achievement facilitated by tools such as the basic perturbation lemma.

Future Prospects in Computational Topology

This research opens avenues for further explorations in AI and computational topology. Extending these algorithms to broader classes of spaces or relaxing constraints like dimensional bounds would revolutionize complexity theory's approach to topological problems. Furthermore, incorporating practical implementations, like leveraging software packages Kenzo and HAP, could democratize access to these advanced computations, fostering innovation across mathematics, computer science, and related disciplines.

In summary, the paper not only enhances algorithmic capabilities in topology but also sets a precedent for addressing historically challenging problems in the field through computational means. Its contributions are set to affect both theoretical explorations and practical applications profoundly.