Image Simple Fold Maps in Topology

Updated 10 July 2026

Image simple fold maps are smooth maps to surfaces where the singular set is embedded, resulting in disjoint closed curves that organize the target's structure.
They stratify the surface into regular regions with controlled fiber transitions, reflecting changes like disk birth, annihilation, or pair-of-pants splits.
Their study bridges singularity theory and topology through Reeb spaces, bubbling operations, and algebraic constraints that link source manifold invariants.

Image simple fold maps are smooth maps to surfaces for which the singular image is embedded. In the surface-target setting, if $M$ is a closed smooth manifold of dimension $m \ge 2$ , $\Sigma$ is a smooth surface, and $f:M\to \Sigma$ , then $f$ is image simple when the restriction $f|_{S(f)}:S(f)\to \Sigma$ is a topological embedding; for fold maps this means the singular image $f(S(f))$ is a finite union of pairwise disjoint embedded closed curves (Saeki et al., 4 Sep 2025). In adjacent parts of the fold-map literature, the same image-level rigidity appears under the conditions that $q_f|_{S(f)}$ is injective, that $f|_{S(f)}$ is an embedding, or that the singular value set is a concentric family of embedded circles or spheres; round fold maps and strongly simple fold maps are standard subclasses of this type (Kitazawa et al., 2021).

For maps to surfaces, the singular set is

$S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$

A point $m \ge 2$ 0 is a fold singularity if, in suitable local coordinates near $m \ge 2$ 1 and $m \ge 2$ 2,

$m \ge 2$ 3

with $m \ge 2$ 4; $m \ge 2$ 5 or $m \ge 2$ 6 gives a definite fold, and the remaining cases are indefinite folds (Saeki et al., 4 Sep 2025). In the $m \ge 2$ 7 case, the local normal forms are particularly concrete: $m \ge 2$ 8 for definite and indefinite folds respectively (Kitazawa et al., 2021).

The singular set of a fold map is a smooth closed $m \ge 2$ 9-dimensional submanifold, and the restriction to the singular set is an immersion; for maps to surfaces this makes $\Sigma$ 0 a 1-dimensional submanifold, hence for closed source manifolds a finite disjoint union of circles (Kitazawa, 2021). For generic maps into surfaces, singularities are folds and cusps, but the present theory focuses on fold maps as endpoint objects even when homotopies may transiently introduce cusps (Saeki et al., 4 Sep 2025).

Several standard classes organize the subject.

Class	Defining condition appearing in the literature	Singular image
Image simple	$\Sigma$ 1 is a topological embedding	Pairwise disjoint embedded closed curves
Simple fold map	$\Sigma$ 2 is injective; equivalently each connected component of each fiber contains at most one singular point	No self-identification in the Reeb space
Strongly simple fold map	$\Sigma$ 3 is an embedding	Embedded 1-submanifold
Round fold map	$\Sigma$ 4 is an embedding and $\Sigma$ 5 is a concentric family of simple closed curves or spheres	Concentric circles or spheres
Directed round fold map	All critical circles are inward-directed	Concentric circles with monotone inward fiber count
Special generic map	All fold singularities have index $\Sigma$ 6	Definite-fold image only

The class of round fold maps is especially rigid: every round fold map is a simple stable map, and in the plane-target case its singular value set is a concentric family of embedded circles, with the outermost circle consisting of definite fold points (Kitazawa et al., 2021). This suggests an image-centered hierarchy in which image simple maps sit between general fold maps and the more structured round and special generic classes.

2. Singular image geometry and fiber change

For image simple fold maps, the embedded singular image stratifies the target surface into regular regions. In the $\Sigma$ 7 round case, after a diffeomorphism of $\Sigma$ 8, one may assume

$\Sigma$ 9

where $f:M\to \Sigma$ 0 is the circle of radius $f:M\to \Sigma$ 1 centered at $f:M\to \Sigma$ 2, and the plane decomposes into the critical value set, annular regular-value regions $f:M\to \Sigma$ 3, the innermost disk, and the outermost complementary annulus (Kitazawa et al., 2021). Crossing a critical circle changes the number of connected components of a regular fiber by exactly one. More precisely, if $f:M\to \Sigma$ 4 meets a critical circle transversely once, then $f:M\to \Sigma$ 5 is a compact surface on which the restriction is a Morse function with exactly one critical point, and

$f:M\to \Sigma$ 6

(Kitazawa et al., 2021).

The distinction between definite and indefinite folds is reflected in this fiber transition. Definite folds create or annihilate a disk component in the fiber, whereas indefinite folds implement a pair-of-pants change: one circle component splits into two, or two merge into one (Kitazawa et al., 2021). In the Reeb-space language, definite folds contribute boundary-type edges, while indefinite folds generate the $f:M\to \Sigma$ 7-shaped branching locus (Kitazawa, 2021).

For round fold maps, the singular circles carry a canonical normal orientation determined by the sign of the change in $f:M\to \Sigma$ 8 of the regular fiber across the circle. A round fold map is directed if all components are inward-directed. If $f:M\to \Sigma$ 9 are the concentric critical circles and $f$ 0 for a regular value $f$ 1 with $f$ 2, then in the directed case $f$ 3 increases by $f$ 4 each time $f$ 5 crosses a critical radius and is strictly increasing toward the central region (Kitazawa et al., 2021). The image thus records not only where singularities occur, but also the monotone or non-monotone evolution of fiber connectivity.

This image-level perspective extends beyond concentric geometry. For simple fold maps into the plane, the singular image is still a 1-dimensional immersed or embedded submanifold, and the intuitive rule remains that definite folds change the number of circle components by birth/death while indefinite folds merge or split components (Kitazawa, 2021). The embeddedness condition suppresses self-intersections among singular value curves and makes the complementary regions and their fiber data combinatorially accessible.

3. Reeb spaces, branched surfaces, and shadow-type models

The Reeb space of a map $f$ 6 is

$f$ 7

where $f$ 8 if $f$ 9 and $f|_{S(f)}:S(f)\to \Sigma$ 0 lie in the same connected component of a fiber $f|_{S(f)}:S(f)\to \Sigma$ 1 for some $f|_{S(f)}:S(f)\to \Sigma$ 2. Writing $f|_{S(f)}:S(f)\to \Sigma$ 3 for the quotient map, there is an induced map $f|_{S(f)}:S(f)\to \Sigma$ 4 with $f|_{S(f)}:S(f)\to \Sigma$ 5 (Kitazawa, 2021). For fold maps, $f|_{S(f)}:S(f)\to \Sigma$ 6 is a polyhedron; for stable fold maps into a surface it is a simple polyhedron, and for simple fold maps it becomes a branched-surface-type object without vertices (Kitazawa, 2021).

For simple fold maps $f|_{S(f)}:S(f)\to \Sigma$ 7, the local model of $f|_{S(f)}:S(f)\to \Sigma$ 8 is particularly explicit. Away from $f|_{S(f)}:S(f)\to \Sigma$ 9, the Reeb space is a 2-manifold and $f(S(f))$ 0 is an immersion. Along $f(S(f))$ 1, each connected component has a bundle neighborhood with fiber either an interval or the $f(S(f))$ 2-shaped 1-dimensional polyhedron

$f(S(f))$ 3

consisting of three unit rays meeting at $f(S(f))$ 4 (Kitazawa, 2021). In normal simple fold maps, these $f(S(f))$ 5-bundles are trivial. This is the low-dimensional source of the “simple polyhedron without vertices” description.

The branched-surface interpretation sharpens the image-level viewpoint. A branched surface is a compact 2-dimensional polyhedron with branch circles whose neighborhoods are either $f(S(f))$ 6 or trivial/ $f(S(f))$ 7-twisted $f(S(f))$ 8-bundles; for SSNS fold maps, the Reeb space is such a branched surface, and the induced map $f(S(f))$ 9 is an immersion away from the branch locus and locally a product map near it (Kitazawa, 2021). In the normal case the $q_f|_{S(f)}$ 0-bundles are trivial, and maps “born from SSNS fold maps” extend the class of immersions of compact surfaces into surfaces by allowing branching at the Reeb-space level rather than cusp singularities in the map between surfaces.

This polyhedral structure admits direct geometric realization. Every graph manifold admits a normal simple fold map $q_f|_{S(f)}$ 1 such that $q_f|_{S(f)}$ 2 is PL-homeomorphic to a 2-dimensional polyhedron assembled by gluing standard disk and pair-of-pants blocks and such that $q_f|_{S(f)}$ 3 is the composition of a PL embedding $q_f|_{S(f)}$ 4 with the orthogonal projection $q_f|_{S(f)}$ 5 (Kitazawa, 2021). In Turaev’s language, $q_f|_{S(f)}$ 6 functions as an un-decorated shadow. The embeddability of such Reeb spaces in $q_f|_{S(f)}$ 7 or in closed orientable 3-manifolds of Heegaard genus $q_f|_{S(f)}$ 8 yields invariants of graph manifolds via the minimal genus of an ambient 3-manifold admitting a PL embedding of $q_f|_{S(f)}$ 9 (Kitazawa, 2021).

A plausible implication is that “image simplicity” is best viewed not merely as embedded critical values in the target surface, but as a condition that makes the quotient geometry of the map finitely stratified, PL-controlled, and directly comparable with the JSJ-type decomposition of the source manifold.

4. Round fold maps and the topology of 3-manifolds

In dimension $f|_{S(f)}$ 0, round fold maps provide a rigid image-simple model with strong topological consequences. A closed orientable 3-manifold admits a round fold map into the plane if and only if it is a graph manifold (Kitazawa et al., 2021). This generalizes the earlier characterization of graph manifolds by simple stable maps into the plane. Since every round fold map is a simple stable map and since the singular value set is a concentric family of embedded circles, the image determines a layered decomposition of the source into pieces diffeomorphic to $f|_{S(f)}$ 1, $f|_{S(f)}$ 2, or $f|_{S(f)}$ 3, where $f|_{S(f)}$ 4 is the pair of pants and $f|_{S(f)}$ 5 (Kitazawa et al., 2021).

The directed subclass is more restrictive. A closed connected orientable graph 3-manifold admits a directed round fold map into $f|_{S(f)}$ 6 if and only if it can be decomposed into finitely many copies of $f|_{S(f)}$ 7 and solid tori $f|_{S(f)}$ 8 such that the corresponding gluing graph is a tree (Kitazawa et al., 2021). An obstruction is the appearance of a non-trivial $f|_{S(f)}$ 9-bundle over $S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 0 among the annular-region pieces in the preimage; such a piece forces alternating inward and outward directions among critical circles. Equivalently, if the normal form plumbing graph contains a loop, then the manifold admits round fold maps but no directed round fold maps (Kitazawa et al., 2021).

The image geometry has further consequences. Non-graph manifolds do not admit round fold maps; in particular, hyperbolic 3-manifolds admit neither round fold maps nor simple stable maps into $S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 1 (Kitazawa et al., 2021). By contrast, every closed orientable Seifert 3-manifold over $S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 2 admits a directed round fold map because it admits a decomposition into $S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 3 and $S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 4 whose graph is a tree (Kitazawa et al., 2021).

Round fold maps also induce open-book structures. If $S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 5 is the closed disk of radius $S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 6, then $S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 7 is an oriented link, and

$S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 8

where $S(f)=\{p\in M\mid \operatorname{rank}(df_p)<2\}.$ 9 is radial projection, is a smooth fiber bundle; thus $m \ge 2$ 00 is a fibered link and $m \ge 2$ 01 admits an open book with binding $m \ge 2$ 02 (Kitazawa et al., 2021). A page $m \ge 2$ 03, for $m \ge 2$ 04, carries a Morse function with exactly $m \ge 2$ 05 critical points, one per critical circle. This gives a direct dictionary between the concentric image in the plane and the open-book/Morse data on the 3-manifold.

The same 3-manifold picture can be restated in Reeb-space terms. For graph manifolds, normal simple fold maps and topologically quasi-trivial round fold maps characterize the manifold, and planarity of the representation graph is equivalent to PL embeddability of the Reeb space of a suitable round fold map in $m \ge 2$ 06 (Kitazawa, 2021). This suggests that, in dimension three, image simple fold maps form a bridge between singularity theory, JSJ-type decomposition, and shadow topology.

5. Construction methods, surgery operations, and bounding manifolds

Several explicit construction mechanisms generate image-simple fold maps while preserving control of the singular image.

A basic surgery framework is given by normal $m \ge 2$ 07-bubbling and $m \ge 2$ 08-bubbling operations. If $m \ge 2$ 09 is obtained from a fold map $m \ge 2$ 10 by normal $m \ge 2$ 11-bubbling along a generating manifold $m \ge 2$ 12 of dimension $m \ge 2$ 13, then for any PID $m \ge 2$ 14,

$m \ge 2$ 15

and

$m \ge 2$ 16

(Kitazawa, 2015). In image terms, the singular value set increases by $m \ge 2$ 17, locally adding a shell around $m \ge 2$ 18. When one starts from a simple fold map whose regular fibers are disjoint unions of almost-spheres, normal $m \ge 2$ 19-bubbling preserves this property and preserves simplicity (Kitazawa, 2015). In the round setting, point bubbling adds a new concentric spherical component to the singular value set.

Another systematic method is the $m \ge 2$ 20-operation on round fold maps. If $m \ge 2$ 21 is a $m \ge 2$ 22 $m \ge 2$ 23-trivial round fold map and $m \ge 2$ 24 is the total space of an $m \ge 2$ 25-bundle over $m \ge 2$ 26 that is trivial over the preimages of small tubular neighborhoods of the singular spheres, then there exists a $m \ge 2$ 27 $m \ge 2$ 28-trivial round fold map $m \ge 2$ 29 (Kitazawa, 2013). For $m \ge 2$ 30, this constructs round fold maps on total spaces of $m \ge 2$ 31-bundles over manifolds already carrying round fold maps. The resulting maps are image-simple in the sense that $m \ge 2$ 32 remains a disjoint union of embedded concentric spheres and $m \ge 2$ 33 is an embedding (Kitazawa, 2013).

Composition with canonical projections yields another source of examples. Special generic maps and certain round fold maps, when composed with the projection $m \ge 2$ 34, often produce round fold maps whose singular value sets are again concentric and whose regular fibers are explicitly described in terms of spheres and connected sums of sphere products (Kitazawa, 2021). This is especially effective when the Reeb space $m \ge 2$ 35 is a disc, a product $m \ge 2$ 36, or a boundary connected sum of elementary blocks.

Image-simple fold maps also support bounding-manifold constructions. Under topologically trivial monodromy on the singular and regular parts and suitable boundaryability assumptions on the fiber data, a simple fold map $m \ge 2$ 37 yields a compact $m \ge 2$ 38-manifold $m \ge 2$ 39 with $m \ge 2$ 40, together with a lower-dimensional polyhedron $m \ge 2$ 41 and simplicial maps $m \ge 2$ 42, $m \ge 2$ 43 such that $m \ge 2$ 44 collapses to a subpolyhedron PL-homeomorphic to $m \ge 2$ 45 (Kitazawa, 2013). In the sphere-fiber case of simple fold maps $m \ge 2$ 46, the resulting $m \ge 2$ 47-manifold $m \ge 2$ 48 collapses onto the Reeb space and the induced map on $m \ge 2$ 49 and $m \ge 2$ 50 is an isomorphism (Kitazawa, 2013).

These construction methods show that image simplicity is compatible with both local surgery and global assembly. The singular image can be enlarged by controlled shells, lifted through bundle constructions, projected from higher-dimensional models, or used as the organizing skeleton of a manifold-with-boundary built from the Reeb space.

6. Homotopy invariants, algebraic constraints, and obstructions

A central image-level invariant is the parity of the number of connected components of the singular set. For an image simple fold map $m \ge 2$ 51 with closed source manifold,

$m \ge 2$ 52

and one considers $m \ge 2$ 53 (Saeki et al., 4 Sep 2025). This parity behaves differently in odd and even source dimensions.

For closed odd-dimensional manifolds of dimension at least $m \ge 2$ 54, parity is not a homotopy invariant. Given any image simple fold map $m \ge 2$ 55, there exists a homotopy through smooth maps to another image simple fold map $m \ge 2$ 56 such that

$m \ge 2$ 57

and the homotopy can be supported in an arbitrarily small disc in the source (Saeki et al., 4 Sep 2025). Two constructive proofs are given: one uses open book decompositions and round fold maps, and the other uses allowable local moves. In the explicit odd-dimensional constructions, the singular set of the homotopy contains a Möbius band component and the restriction to the singular set has a triple self-intersection point (Saeki et al., 4 Sep 2025).

For closed even-dimensional source manifolds and orientable targets, parity is a homotopy invariant. The proof uses Euler characteristic accounting over the complementary regions of the embedded multicurve $m \ge 2$ 58: when one crosses a fold curve, the Euler characteristic of the regular fiber changes by $m \ge 2$ 59, and the parity of $m \ge 2$ 60 is determined by $m \ge 2$ 61 and $m \ge 2$ 62 modulo $m \ge 2$ 63 (Saeki et al., 4 Sep 2025). This fails for non-orientable targets: for every $m \ge 2$ 64, there exist a closed non-orientable $m \ge 2$ 65-manifold and two homotopic image simple fold maps into the open Möbius band with different parity of the number of fold components (Saeki et al., 4 Sep 2025).

Image-simple and simple fold maps also impose homotopical and cohomological constraints through their Reeb spaces. If $m \ge 2$ 66 is a simple fold map whose singular indices are $m \ge 2$ 67 or $m \ge 2$ 68 and whose regular fibers are disjoint unions of almost-spheres, then for any ring $m \ge 2$ 69 and $m \ge 2$ 70,

$m \ge 2$ 71

(Kitazawa, 2015). Thus low-degree algebraic topology of the source is encoded by the image-level quotient.

For special generic maps, the Reeb space is even more restrictive: $m \ge 2$ 72 is a smooth $m \ge 2$ 73-manifold immersed in the target, and one obtains vanishing theorems for cup products and triple Massey products in ranges controlled by the collapse dimension of $m \ge 2$ 74 (Kitazawa, 2020). In particular, a closed simply-connected 7-manifold with a nonvanishing triple Massey product in degree $m \ge 2$ 75 admits no special generic map into $m \ge 2$ 76 for $m \ge 2$ 77 (Kitazawa, 2020). In the 3-manifold round setting, directedness imposes an additional homological obstruction: if a closed orientable 3-manifold admits a directed round fold map into $m \ge 2$ 78, then for every $m \ge 2$ 79, the cup product $m \ge 2$ 80 vanishes in $m \ge 2$ 81 (Kitazawa et al., 2021).

Current developments indicate two complementary themes. One is rigidity: embedded singular images force strong restrictions on Reeb spaces, plumbing graphs, open books, and algebraic invariants. The other is flexibility: bubbling operations, $m \ge 2$ 82-operations, projection constructions, and allowable moves permit controlled modification of the singular image, often by adding new embedded components. This suggests that image simple fold maps occupy a technically precise middle ground between arbitrary generic maps to surfaces and the highly structured subclasses of special generic and round fold maps.