On Computability and Triviality of Well Groups

Area of Science:

• Algebraic Topology
• Computational Mathematics
• Differential Geometry

Background:

• Well groups capture robust homological properties of zero sets of continuous maps.
• Existing methods for well group computation are limited, particularly for general cases.
• The invariance of well groups under small perturbations is crucial but their computability is restricted.

Purpose of the Study:

• To improve the computability of well groups for continuous maps on compact spaces.
• To investigate the limitations of well groups as invariants by identifying cases where they are incomplete.
• To propose alternative invariants with better descriptive power and computability.

Main Methods:

• Identification of a computable subgroup of the well group using cap product with the pullback of orientation.
• Algorithmic approximation of well groups from below.
• Construction of specific examples of maps to demonstrate the limitations of well groups.

Main Results:

• A computable subgroup of the well group is identified, allowing for algorithmic approximation.
• The approximation is exact for smooth maps when the target space is a sphere.
• Examples are provided where well groups are isomorphic, yet perturbations of the maps yield distinct zero sets, indicating incomplete invariance.

Conclusions:

• Well groups can be algorithmically approximated, enhancing their practical applicability.
• Well groups are not always complete invariants, necessitating the search for more powerful topological invariants.
• Further research is needed to develop new invariants for vector-valued maps that offer improved descriptive power and computability.