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Updated: Jun 2, 2026
16:23
Fiber Connections of the Supplementary Motor Area Revisited: Methodology of Fiber Dissection, DTI, and Three Dimensional Documentation
Published on: May 23, 2017
Fiber-connected, indefinite Morse 2-functions on connected n-manifolds
1Euclid Lab, Iowa City, IA 52245, USA. d.gay@euclidlab.org.
Summary
We introduce Morse 2-functions, which are smooth maps between manifolds and surfaces. Key properties include fiber connectivity and avoiding specific local extrema for applications in smooth invariant theory.
Area of Science:
- Differential Geometry
- Topology
- Mathematical Physics
Background:
- Morse theory is a fundamental tool in topology and geometry.
- Understanding smooth maps between manifolds and surfaces is crucial for developing new mathematical invariants.
- Classical Morse theory analyzes functions on manifolds, but extensions to higher dimensions are needed.
Purpose of the Study:
- To introduce and analyze a new class of mathematical objects called Morse 2-functions.
- To investigate the properties of fiber connectivity and indefiniteness in Morse 2-functions.
- To lay the groundwork for defining novel smooth invariants using these functions.
Main Methods:
- Studying generic smooth maps from smooth manifolds to smooth surfaces.
- Analyzing homotopies between these maps.
- Investigating conditions for fiber connectivity and avoiding local extrema over one-dimensional submanifolds.
Main Results:
- Defined Morse 2-functions as smooth maps from manifolds to surfaces.
- Identified 'fiber-connected' and 'indefinite' as crucial properties of Morse 2-functions.
- Established foundational concepts for using Morse 2-functions in invariant theory.
Conclusions:
- Morse 2-functions offer a new framework for studying smooth maps and their properties.
- The concepts of fiber connectivity and indefiniteness are key to their analysis.
- This work provides a basis for developing advanced smooth invariants analogous to Morse homology and Cerf theory.
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