Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

On a classification of Morse functions on $3$-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A labelled Reeb digraph gives a complete classification of sphere-and-torus-fibered Morse functions on 3-manifolds.

desk verdict A plausible 3D Reeb digraph realization theorem whose converse rests on a load-bearing but explicitly unproved local move; deserves a serious referee. read the letter →

arxiv 2411.15943 v2 pith:NGMKFM2K submitted 2024-11-24 math.GT math.AT

Naoki Kitazawa This is my paper · ORCID
classification math.GTmath.AT MSC 57R4557R19
keywords MorsefunctionsReebgraphsdigraphs3-dimensionalmanifoldsHeegaardgenussphere-torus-fiberedconnectedsumsSSTF
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to classify Morse functions on closed orientable 3-manifolds whose regular level sets are only spheres and tori, by encoding each function in a labelled Reeb digraph. The central theorem says that a digraph with first Betti number $a$, $b$ degree-2 vertices where both incident edges carry the sphere label, and $c$ edges carrying the torus label, is realized by such a Morse function precisely when the domain is a connected sum of at least $a+b$ copies of $S^1 \times S^2$ and at most $c$ Heegaard genus-one summands with finite fundamental group. This gives a higher-dimensional analogue of the known surface classification and pinpoints the topological restriction found earlier for sphere-fibered Morse functions. A sympathetic reader cares because classification of Morse functions in dimensions three and up has largely been open, and this pins down exactly which 3-manifolds admit the simplest possible fiber structure.

What carries the argument

The carrying object is the Reeb digraph of a Morse function: vertices are connected components of level sets that contain critical points, edges correspond to regular level-set components, oriented by increasing value, and each edge is labelled by whether the generic fibre over it is $S^2$ or $S^1 \times S^1$. The proof also uses SSTF Morse functions and the handle-calculus description of how singular points change the level surface. Around each vertex the paper constructs a local SSTF Morse function by attaching 2-handles and 1-handles in prescribed orders, and glues these local models along trivial fibre bundles over the edges. The converse step applies local Reeb graph modifications, obtained by applying the operations of [16] twice, to force the torus-labelled edges into a normal form whose topology can be read off.

What would settle it

A closed orientable 3-manifold with an SSTF Morse function whose torus-labelled Reeb subgraph has a component not meeting an extremal critical value, or has more than $c$ such components, would contradict the converse part of the classification; the paper's Step 3 asserts such components can always be deformed away, so exhibiting a function whose torus-labelled loop cannot be removed would settle the claim.

Watch

Extended reading notes

Core claim

On a closed connected orientable 3-manifold, consider a Morse function for which every regular preimage is a disjoint union of copies of $S^2$ and $S^1 \times S^1$ ('simple sphere-torus-fibered', or SSTF). Theorem 2 asserts that a finite connected digraph $K$, whose edges are labelled by whether the generic fiber is a sphere or a torus, occurs as the Reeb digraph of such a function if and only if the manifold is a connected sum of $r \geq a+b$ copies of $S^1 \times S^2$ and $c' \leq c$ copies of Heegaard genus-one manifolds with finite fundamental group, where $a$ is the first Betti number of $K$, $b$ is the number of degree-2 vertices whose two incident edges are both sphere-labelled, and $c$ is the number of torus-labelled edges. In other words, the labelled graph completely determines the connected-sum type of any 3-manifold carrying the function, and every manifold of that type carries such a function. The proof is constructive in one direction: local models around vertices are glued along trivial bundles over edges, and handle attachments give the required preimages. The converse direction uses a deformation result [20] to simplify the function, then analyses the Reeb digraph to read off the connected-sum splitting.

Load-bearing premise

The converse proof assumes that any such Morse function can be deformed, without changing the manifold, into one whose torus-labelled Reeb components are disjoint closed intervals each meeting an extremal critical value, with at most $c$ components, and the paper notes the local Reeb graph moves behind this are asserted rather than directly proved on 3-manifolds.

Editorial extensions

If this is right

  • Any digraph satisfying the stated numeric conditions is realized by an SSTF Morse function on every sufficiently large connected sum: $r \geq a+b$ copies of $S^1 \times S^2$ and $c' \leq c$ Heegaard genus-one summands with finite fundamental group.
  • Conversely, if a 3-manifold admits an SSTF Morse function with such a Reeb digraph, the manifold must be one of these connected sums, so the digraph data give a diffeomorphism-type obstruction.
  • The case $c=0$ recovers the sphere-fibered classification: such functions exist exactly on connected sums of $S^1 \times S^2$, or on $S^3$, matching the higher-dimensional analogue of the surface theorem.
  • The number of torus-labelled edges $c$ and the count $c'$ of non-spherical summands satisfy $c' \leq c$, so the labelled graph bounds how many nontrivial Heegaard genus-one summands can appear.
  • For a fixed manifold of the allowed form, the construction yields such a Morse function for any $r$ in the allowed range and any choice of $c' \leq c$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The theorem suggests that the minimal possible number of torus-labelled edges in a realizing Reeb digraph is a diffeomorphism invariant of a 3-manifold; the paper's construction gives an upper bound but does not address minimization.
  • The Morse-Bott counterexample described in the remarks shows the analogous statement for functions with critical tori fails, so the classification is specific to genuine Morse functions; checking whether allowing higher-genus regular fibers yields new domain manifolds would test how far the connected-sum restriction extends.
  • One could formulate a computational test: present a 3-manifold by a Heegaard diagram, compute the candidate Reeb graph data $a,b,c$, and check the inequalities $r \geq a+b$ and $c' \leq c$; the theorem predicts exactly which candidates are realizable.
  • The deformation step whose local moves are imported from [16] is the structural subtlety of the converse; if those moves fail on some 3-manifold, the converse would need a weaker statement.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes Theorem 2, a classification/realization statement for Morse functions on closed connected orientable 3-manifolds whose regular level sets are disjoint unions of spheres and tori. Given a finite connected digraph K with first Betti number a, with b vertices of degree 2 whose two incident edges are both sphere-labelled, and with c torus-labelled edges, the theorem asserts that such a graph is realized as the Reeb digraph of a Morse function on any 3-manifold diffeomorphic to a connected sum of r≥a+b copies of S^1×S^2 and c'≤c Heegaard-genus-one manifolds with finite fundamental group; conversely, any Morse function whose Reeb digraph and regular fibers match this description has a domain manifold of this connected-sum form. The proof constructs local Morse models via handle attachments (STEP 2) and then uses local Reeb graph modifications to control the number of torus-labelled edges and thereby the number of Heegaard-genus-one summands (STEP 3). The paper also discusses extensions to Morse-Bott functions and relations to Saeki's theorem and to work of Michalak.

Significance. If Theorem 2 is correct, it gives a genuinely higher-dimensional analogue of the surface Reeb-graph realization theorems of Gelbukh and Michalak, and it sharpens Saeki's sphere-torus fibered Morse function theorem by tying the number of torus-fibred Reeb edges to the number of Heegaard-genus-one connected-summands. The forward direction, with its explicit handle counts and local models, is plausible and demonstrates real technical creativity. However, the converse direction currently rests on an asserted but unproved deformation of Reeb graphs, and the forward construction contains several explicitly admitted gaps ('we do not present them precisely', 'we do not explain this precisely'). The paper is a promising draft rather than a complete proof; its central claim is defensible but needs substantial repair.

major comments (3)
  1. [STEP 3, paragraph beginning 'We can deform the function f0 and the Reeb digraph W_f0...'] The converse of Theorem 2 depends on the assertion that any SSTF Morse function can be deformed so that the torus-labelled subgraph W_{f0,S1×S1} becomes a disjoint union of closed intervals, each meeting an extremal critical value, with at most c components. This is exactly what bounds c'≤c. The text explicitly concedes 'we cannot show this argument on the 3-dimensional manifold from them directly' and instead says the local movements can be realized by applying operations from [16, Figure 5] twice. No proof of the 3-manifold invariance of the Figure 5 replacements is supplied. Since this step converts an arbitrary torus-edge configuration into at most c separated torus fibres, a failure here would allow more Heegaard-genus-one summands than torus-labelled edges and would invalidate the converse. This is a load-bearing gap, not a mere presentation issue.
  2. [STEP 2, subsection (2) and the paragraph after the bullet list] The local model at a vertex of degree 2 whose two incident edges are sphere-labelled is said to have Reeb space of first Betti number k≥1 (with k>0 because singular points must be present). Gluing such local models into the global Reeb graph K, which has first Betti number a, appears to force the first Betti number of the constructed Reeb graph to be at least a+b, and the text later states that the first Betti number of W_{f0} can be made an arbitrary integer r≥a+b. But the final Morse function f must have Reeb digraph isomorphic to K, whose first Betti number is exactly a. The text never explains how the homotopy from f0 to f reduces the Betti number from r (or at least a+b) down to a, or how the extra cycles created at degree-2 vertices are eliminated without changing the manifold. This gap affects the existence direction of Theorem 2.
  3. [STEP 2, last bullet of the bullet list] The sentence 'The resulting 3-dimensional closed and connected manifold is diffeomorphic to the sphere S3 in the case a+b=0' is imprecise: if a+b=0 and r>0, the constructed manifold is a connected sum of r copies of S^1×S^2, not S^3. The theorem allows arbitrary r≥a+b, so the case a+b=0 with r≥1 is not treated by the stated sentence. This is a local inaccuracy, but it obscures the content of the existence statement in exactly the case where the graph has no cycles and no degree-2 vertices.
minor comments (4)
  1. [Theorem 2 statement, condition (4)] The phrase 'whose free groups are finite' should read 'whose fundamental groups are finite'; the word 'free' is a typo that appears twice in the same sentence.
  2. [STEP 2, introductory paragraph] The text repeatedly says 'we do not present them precisely' and 'we do not explain this precisely' in connection with elementary Morse-theoretic handle arguments. These remarks are honest but make it difficult for the reader to verify the local model, in particular the claim that the handle attachments at degree-2 vertices yield exactly the prescribed Reeb graph with the stated edge labels.
  3. [Section 3, Problem 1 paragraph] The counterexample to a Morse-Bott extension of Theorem 3 is described in one sentence with no verification that the resulting Morse-Bott function has the stated preimage structure. Since the example is used to motivate the open problem, a short proof or diagram would improve readability.
  4. [References] The paper cites several of the author's own works ([7], [9], [10], [11]) and relies on [16, Figure 5] as a black box for the critical deformation in STEP 3; the referee recommends that a revision give a self-contained statement of the lemma being imported from [16].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is independent and the converse rests on external theorems; the STEP 3 gap is a proof gap, not a circular reduction.

full rationale

The derivation chain of Theorem 2 is not circular. The forward direction explicitly constructs the desired SSTF Morse function from the digraph by local handle attachments and gluing of trivial bundles over edges, so it does not presuppose the conclusion. The converse is obtained by deforming an arbitrary SSTF Morse function to a normal form and then invoking Saeki's Theorem 3 ([20, Theorem 6.5]) and Michalak's local moves ([16, Figure 5]) as external results; these are outside the paper's own asserted classification, so they are genuine independent support rather than a self-referential loop. The author's self-citations [7,9,10,11] are used only for background on Reeb graphs and are explicitly not assumed: the paper states 'we do not assume related knowledge and arguments at all.' The genuine weakness is in STEP 3, where the claim that the torus-labelled subgraph can be deformed into disjoint closed intervals and the 3-dimensional realizability of the Figure 5 moves is asserted, with the admission 'we cannot show this argument on the 3-dimensional manifold from them directly' and the substitute appeal to [16, Figure 5] twice is not fully demonstrated. That is an omitted proof or rigor gap, not a reduction of the theorem to its own conclusion, so it does not constitute circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard Morse theory and handle decompositions, Reeb graph theory, Saeki's classification of manifolds admitting sphere and torus fibered Morse functions, and 3-manifold Heegaard genus facts. No free parameters are fitted and no new physical or mathematical entities are postulated.

assumptions (5)
  • standard math Morse functions admit handle decompositions, with each critical point corresponding to a handle of appropriate index.
    Used throughout STEP 2 to construct local functions and to count singular points. The paper cites [18] for Morse theory.
  • standard math The Reeb space of a Morse function on a closed manifold with these preimage conditions is a finite graph, and the Reeb digraph structure is well defined.
    Invoked in Section 1 and used to define the objects of Theorem 2; the paper cites [21] for rigour.
  • domain assumption Saeki's Theorem 3 ([20, Theorem 6.5]): a closed orientable 3-manifold admits a Morse function with nonsingular preimages that are disjoint unions of copies of S^2 and S^1×S^1 if and only if it is a connected sum of copies of S^1×S^2 and Heegaard genus 1 manifolds with finite fundamental groups.
    Used as the external classification supporting the converse direction in STEP 3. It is quoted but not proved in this paper.
  • domain assumption Heegaard genus classification: the only genus 0 manifold is S^3, and a Heegaard genus 1 manifold is diffeomorphic to S^2×S^1 or a lens space.
    Used in Section 1 to specify the manifold class in Theorem 2, citing [6].
  • domain assumption Relations between the fundamental group of the manifold and the Reeb graph, as established in [22, Corollary 4.8], [7], and [8, Corollary 4], determine the connected-sum type.
    Used in STEP 2 to identify the diffeomorphism type of the constructed manifold from its Reeb graph and fundamental group.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On a classification of Morse functions on $3$-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one." pith.science (2026). https://pith.science/paper/NGMKFM2K

@misc{pith2026241115943,
  author       = {Pith},
  title        = {Pith review of: On a classification of Morse functions on $3$-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGMKFM2K}},
  note         = {Machine review of arXiv:2411.15943}
}
abstract

Morse functions are important objects and tools in understanding topologies of manifolds since the 20th century. Their classification has been natural and difficult problems, and surprisingly, this is recently developing. Since the 2010's, results for cases of surfaces have been presented by Gelbukh, Marzantowicz and Michalak for example. We have also longed for higher dimensional cases. We present a classification of Morse functions on $3$-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one. We concentrate on Morse functions such that preimages of single points containing no singular points are disjoint unions of spheres and tori. Existence of such functions implies that the $3$-dimensional closed and connected manifolds are of such manifolds. This has been shown by Saeki in 2006 and we further study structures of these functions.

Figures

Figures reproduced from arXiv: 2411.15943 by the authors.

Figure 1
Figure 1. Local information on the Reeb (di)graph and preimages for SSTF Morse functions. A blue (red) edge shows an edge at a point in the interior of which the preimage is diffeomorphic to the sphere S 2 (resp. the torus S 1 × S 1 ). A black (green) dot is for a singular point of index 0 (resp. 1) for the Morse function. c ′−1 (R − (q − ǫ, q + ǫ)) in such a way that at all singular points of c, the values are same and q. Ot… view at source ↗
Figure 2
Figure 2. The Reeb digraph Wf0,v for k = 2. The preimage of a point in the interior of a blue colored edge is diffeomorphic to S 2 . Green dots are for singular points of index 1 or 2 for the Morse function. More precisely, we choose a 2-dimensional disk D2 smooothly embed￾ded in the interior of each connected manifold which is a 2-dimensional disk and represented as the intersection of each of the k1 + 1 spheres from Sv,1 an… view at source ↗
Figure 3
Figure 3. The Reeb digraph Wf0,v for iv,S2 = iv,S1×S1 = ov,S2 = ov,S1×S1 = 1. The preimage of a point in the interior of a blue (red) colored edge is diffeomorphic to S 2 (resp. S 1 × S 1 ). Green dots are for singular points of index 1 or 2 for the function. iv,S2 +iv,S1×S1 −2 of all iv,S2 +iv,S1×S1 disks before, we add an￾other smoothly embedded 2-dimensional disk D2 in its interior disjointly from the previously chosen dis… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: An example of attachment of handles to the sur￾face f0,v −1 (tv,1 + ǫ), colored in gray. This is also for the case iv,S2 = iv,S1×S1 = ov,S2 = ov,S1×S1 = 1 or FIGURE 3. The mani￾fold Dv,1,1, diffeomorphic to S 1 × D1 , is colored in black, the disks Dv,2,1 and Dv,2,2 ar…
Figure 5
Figure 5. Figure 5: Important local changes of Reeb digraphs. Colors are used as in the previous Figures. Of course for these local SSTF functions fv, we can consider −fv and have similar cases. (2) For each connected component Iv ⊂ Wf0,S1×S1 before, represented as a closed interval, the …

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions

    math.GT 2024-12 conditional novelty 6.0 of 10

    A closed orientable 3-manifold admits such a Morse-Bott function exactly when it is a connected sum of lens spaces, copies of S2 × S1, and torus bundles over S1.

Reference graph

Works this paper leans on

23 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [17]

    L. P. Michalak, Reeb graph invariants of Morse functions and 3-manifold groups , arXiv:2403.02291

    arXiv

  2. [16]

    L. P. Michalak, Combinatorial modifications of Reeb graphs and the realizat ion problem , Discrete Comput. Geom. 65 (2021), 1038–1060, arXiv:1811.0 8031

    2021

  3. [1]

    Gelbukh, A finite graph is homeomorphic to the Reeb graph of a Morse-Bot t function , Mathematica Slovaca, 71 (3), 757–772, 2021; doi: 10.1515/m s-2021-0018

    I. Gelbukh, A finite graph is homeomorphic to the Reeb graph of a Morse-Bot t function , Mathematica Slovaca, 71 (3), 757–772, 2021; doi: 10.1515/m s-2021-0018

    doi:10.1515/m 2021

  4. [2]

    Gelbukh, Morse-Bott functions with two critical values on a surface , Czechoslovak Mathe- matical Journal, 71 (3), 865–880, 2021; doi: 10.21136/CMJ

    I. Gelbukh, Morse-Bott functions with two critical values on a surface , Czechoslovak Mathe- matical Journal, 71 (3), 865–880, 2021; doi: 10.21136/CMJ. 2021.0125-20

    arXiv 2021

  5. [3]

    Gelbukh, Criterion for a graph to admit a good orientation in terms of l eaf blocks, Monatsh

    I. Gelbukh, Criterion for a graph to admit a good orientation in terms of l eaf blocks, Monatsh. Math., 198, 61–77, 2022

    2022

  6. [4]

    Gelbukh, Realization of a digraph as the Reeb graph of a Morse-Bott fun ction on a given surface, Topology and its Applications, 2024

    I. Gelbukh, Realization of a digraph as the Reeb graph of a Morse-Bott fun ction on a given surface, Topology and its Applications, 2024

    2024

  7. [5]

    Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024

    I. Gelbukh, Reeb Graphs of Morse-Bott Functions on a Given Surface , Bulletin of the Iranian Mathematical Society, Volume 50 Article number 84, 2024

    2024

  8. [6]

    Hempel, 3- Manifolds , AMS Chelsea Publishing, 2004

    J. Hempel, 3- Manifolds , AMS Chelsea Publishing, 2004

    2004

Show all 23 references
  1. [7]

    Kitazawa, On manifolds admitting fold maps with singular value sets of concentric spheres, Doctoral Dissertation, Tokyo Institute of Technology (201 4)

    N. Kitazawa, On manifolds admitting fold maps with singular value sets of concentric spheres, Doctoral Dissertation, Tokyo Institute of Technology (201 4)

  2. [8]

    Kitazawa, Fold maps with singular value sets of concentric spheres , Hokkaido Mathemat- ical Journal Vol.43, No.3 (2014), 327–359

    N. Kitazawa, Fold maps with singular value sets of concentric spheres , Hokkaido Mathemat- ical Journal Vol.43, No.3 (2014), 327–359

    2014

  3. [9]

    Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed ori- entable manifolds with finitely many singular values , Topol

    N. Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed ori- entable manifolds with finitely many singular values , Topol. Methods in Nonlinear Anal. Vol. 59 No. 2B, 897–912, arXiv:1902.08841

    1902 arXiv

  4. [10]

    Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed mani- folds which may not be orientable , Methods of Functional Analysis and Topology Vol

    N. Kitazawa, On Reeb graphs induced from smooth functions on 3-dimensional closed mani- folds which may not be orientable , Methods of Functional Analysis and Topology Vol. 29 No. 1 (2023), 57–72, 2024

    2023

  5. [11]

    N. Kitazawa, Realization problems of graphs as Reeb graphs of Morse funct ions with pre- scribed preimages, the 2nd revised version is submitted to a refereed journal b ased on positive comments, arXiv:2108.06913

    arXiv

  6. [12]

    Martinez-Alfaro, I

    J. Martinez-Alfaro, I. S. Meza-Sarmiento and R. Olivei ra, Topological classification of simple Morse Bott functions on surfaces , Contemp. Math. 675 (2016), 165–179

    2016

  7. [13]

    Marzantowicz and L

    W. Marzantowicz and L. P. Michalak, Relations between Reeb graphs, systems of hypersur- faces and epimorphisms onto free groups , Fund. Math., 265 (2), 97–140, 2024

    2024

  8. [14]

    Masumoto and O

    Y. Masumoto and O. Saeki, A smooth function on a manifold with given Reeb graph , Kyushu J. Math. 65 (2011), 75–84

    2011

  9. [15]

    L. P. Michalak, Realization of a graph as the Reeb graph of a Morse function on a manifold . Topol. Methods in Nonlinear Anal. 52 (2) (2018), 749–762, ar Xiv:1805.06727

    2018 arXiv

  10. [18]

    Milnor, Lectures on the h-cobordism theorem , Math

    J. Milnor, Lectures on the h-cobordism theorem , Math. Notes, Princeton Univ. Press, Prince- ton, N.J. 1965

    1965

  11. [19]

    G. Reeb, Sur les points singuliers d´une forme de Pfaff compl´ etement int` egrable ou d´une fonction num´ erique, Comptes Rendus Hebdomadaires des S´ eances de I´Acad´ emiedes Sciences 222 (1946), 847–849

    1946

  12. [20]

    Saeki, Morse functions with sphere fibers , Hiroshima Math

    O. Saeki, Morse functions with sphere fibers , Hiroshima Math. J. Volume 36, Number 1 (2006), 141–170

    2006

  13. [21]

    O. Saeki, Reeb spaces of smooth functions on manifolds , International Mathe- matics Research Notices, maa301, Volume 2022, Issue 11, Jun e 2022, 3740–3768, https://doi.org/10.1093/imrn/maa301, arXiv:2006.0168 9

    2022

  14. [22]

    Saeki and K

    O. Saeki and K. Suzuoka, Generic smooth maps with sphere fibers J. Math. Soc. Japan Volume 57, Number 3 (2005), 881–902

    2005

  15. [23]

    Sharko, About Kronrod-Reeb graph of a function on a manifold , Methods of Functional Analysis and Topology 12 (2006), 389–396

    V. Sharko, About Kronrod-Reeb graph of a function on a manifold , Methods of Functional Analysis and Topology 12 (2006), 389–396. Institute of Mathematics for Industry, Kyushu University, 7 44 Motooka, Nishi-ku Fukuoka 819-0395, Japan, TEL (Office): +81-92-802-4402, F AX (Offi...

    2006

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.