REVIEW 7 cited by
On Reeb graphs induced from smooth functions on 3-dimensional closed orientable manifolds with finitely many singular values
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
abstract
The Reeb graph of a function on a smooth manifold is the graph obtained as the space of all connected components of level sets such that the set of all vertices coincides with the set of all connected components of level sets including singular points. Reeb graphs are fundamental and important in the algebraic and differential topological theory of Morse functions and their generalizations. In this paper, as a related fundamental and important study, for given graphs, we construct certain smooth functions inducing the graphs as the Reeb graphs. Such works have been demonstrated by Masumoto, Michalak, Saeki, Sharko etc. and also by the author since 2000s. We present new smooth functions on suitable $3$-dimensional closed orientable manifolds through explicit constructive methods.
Forward citations
Cited by 7 Pith papers
-
Arrangements of small circles for Morse-Bott functions
The paper classifies the local changes to Poincaré-Reeb graphs caused by adding small circles centered on existing circles, for circle arrangements associated with Morse-Bott functions.
-
Compactifying real analytic functions and resulting Reeb spaces
Explicit real-analytic projections on manifolds in Euclidean space realize prescribed normal graph diagrams for NF of their Reeb digraph compactifications.
-
Fundamental examples of height functions on closed manifolds and their 1st derivatives
For closed manifolds obtained by mixing two spheres and a real curve, the paper classifies the critical sets of the height function and of its newly introduced first-derivative function.
-
Regions represented as foliated forms and natural smooth maps onto them
For certain smoothly foliated regions, explicit submanifolds are built whose first-coordinate projection has no critical points, and the critical set of the induced derivative is computed in an explicit two-dimensiona...
-
Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions
Every tree, and certain graphs assembled from single edges and small circles, is the Reeb graph of a Morse-Bott real algebraic function defined by degree-1 and degree-2 polynomials.
-
Reconstruction of real algebraic functions into curves with prescribed Reeb graphs
For any finite graph satisfying genericity conditions and any dimension at least 2, the paper constructs real algebraic maps to curves whose Reeb graph is isomorphic to the graph.
-
On a classification of Morse functions on $3$-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one
For 3-manifolds made by connected sums of S^1×S^2 and lens spaces, Morse functions whose nonsingular level sets are spheres and tori are classified by their Reeb graphs with sphere or torus labels on each edge.
Discussion (0). Continue with ORCID to comment.