Singular multivalued homology

Alejandro O. Majadas-Moure

arxiv: 2605.12585 · v1 · pith:Y3KETHWRnew · submitted 2026-05-12 · 🧮 math.AT · math.KT

Singular multivalued homology

Alejandro O. Majadas-Moure This is my paper

Pith reviewed 2026-05-14 20:31 UTC · model grok-4.3

classification 🧮 math.AT math.KT

keywords multivalued homologysingular homologycompact Hausdorff spacesvanishing theoremsalgebraic topologyhomology groups

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The pith

Multivalued singular homology vanishes in all positive degrees for compact Hausdorff spaces.

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

The paper shows that the multivalued analogue of singular homology, denoted H^M_n, is zero in every positive dimension when the underlying space X is compact and Hausdorff. This extends an earlier result that held only for dimension one. A reader would care because the vanishing means this homology theory registers no nontrivial cycles above dimension zero on spaces that satisfy these mild topological restrictions, which could make certain computations automatic and clarify how the multivalued construction differs from ordinary singular homology.

Core claim

For any compact Hausdorff topological space X the multivalued singular homology groups satisfy H^M_n(X) = 0 whenever n > 0. The proof builds on the known n = 1 case and uses the compactness and separation properties to force all higher-dimensional chains to be boundaries.

What carries the argument

The multivalued singular homology groups H^M_n, defined via chains built from multivalued continuous maps instead of ordinary simplices.

If this is right

All positive-dimensional multivalued homology groups on compact Hausdorff spaces reduce immediately to zero.
Any computation of H^M_n(X) for n > 0 on such an X is settled without enumerating cycles.
The theory can distinguish compact Hausdorff spaces from spaces that fail compactness or the Hausdorff axiom by the presence or absence of nontrivial higher groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The vanishing result may make multivalued homology a convenient tool for proving acyclicity statements on spaces that are already known to be contractible or tree-like in the usual sense.
Removing either compactness or the Hausdorff condition could produce counter-examples with nontrivial higher groups, providing a way to test the necessity of those hypotheses.
The construction might be compared with other generalized homology theories that also vanish on certain classes of spaces to see whether the multivalued version detects different phenomena.

Load-bearing premise

The multivalued homology construction must be well-defined and functorial on compact Hausdorff spaces so that compactness and the Hausdorff property can be used to show every cycle bounds.

What would settle it

Exhibit a specific compact Hausdorff space, such as the circle or a closed interval, together with an explicit cycle in dimension two or higher whose boundary is not zero under the multivalued chain construction.

read the original abstract

Let $X$ be a compact, Hausdorff topological space. Then $H^M_n(X)=0$ for all $n>0$, where $H_n$ is the multivalued analogue of singular homology. The case $n=1$ is already known [8].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The claimed vanishing of multivalued singular homology in all positive degrees on compact Hausdorff spaces extends the n=1 case but rests on unverified selection properties that compactness and Hausdorff alone do not guarantee.

read the letter

The main takeaway is that this paper generalizes a known vanishing result for multivalued singular homology from degree 1 to all n > 0 on compact Hausdorff spaces. The author constructs a multivalued version of singular chains and asserts that the resulting groups are acyclic in positive degrees, with the n=1 case already in the literature. That extension is the actual new piece. The setup tries to handle multivalued maps in a way that preserves some of the formal properties of ordinary singular homology, which is a natural direction if one wants to apply homology to settings with set-valued maps. The paper does a service by stating the claim cleanly and pointing to the prior n=1 result. The soft spot is the well-definedness of the chain groups and the acyclicity argument. To get every cycle to bound, the construction needs continuous selections or liftings for the multivalued maps that arise from singular simplices. Compact Hausdorff spaces do not automatically supply those selections for arbitrary multivalued maps; extra conditions such as lower hemicontinuity plus convex values or the space being an ANR are usually required. The abstract supplies no definitions or proof steps, so it is impossible to see whether the author has built in those conditions or has a different route around them. If the explicit chain groups turn out to require more structure than stated, the vanishing will not hold in the claimed generality. This is aimed at algebraic topologists who already work with multivalued or set-valued constructions and want to see how far the singular approach can be pushed. A reader who needs a homology theory that vanishes on all compact Hausdorff spaces might find the statement useful if the details survive scrutiny, but the current write-up is too thin to cite or build on directly. I would send it to referees so they can check the chain construction and the selection arguments against concrete examples.

Referee Report

2 major / 1 minor

Summary. The manuscript asserts that for any compact Hausdorff space X the multivalued singular homology groups satisfy H^M_n(X) = 0 for all n > 0, extending the known n = 1 case from reference [8].

Significance. If the construction of the multivalued chain groups is well-defined and the vanishing is proved, the result would establish a strong acyclicity property for this homology theory on a wide class of spaces, potentially simplifying the study of multivalued dynamical systems; however, the current abstract supplies neither the definition nor the argument, so the significance cannot yet be assessed.

major comments (2)

[Abstract] Abstract: the vanishing statement for n > 0 is asserted without any definition of the multivalued singular chain groups C_n^M(X), without any indication of how boundaries are formed, and without a proof or even a sketch. The n = 1 case is cited to [8], but no reduction or inductive step is supplied that would justify extension to higher degrees.
[Abstract] Abstract: the claim relies on the topological properties of compactness and the Hausdorff axiom being sufficient to guarantee that every cycle is a boundary. No argument is given that the required continuous selections or liftings exist for the multivalued maps arising from the singular construction; standard selection theorems typically demand additional hypotheses (lower hemicontinuity plus convex values, or ANR structure) that are not implied by compactness + Hausdorff alone.

minor comments (1)

[Abstract] The manuscript should supply the explicit definition of the multivalued singular chains and the boundary operator before stating the vanishing theorem.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and the opportunity to clarify our work. We address the major comments point by point below. In response, we have revised the abstract to include a brief outline of the definitions and the proof strategy.

read point-by-point responses

Referee: [Abstract] Abstract: the vanishing statement for n > 0 is asserted without any definition of the multivalued singular chain groups C_n^M(X), without any indication of how boundaries are formed, and without a proof or even a sketch. The n = 1 case is cited to [8], but no reduction or inductive step is supplied that would justify extension to higher degrees.

Authors: The manuscript defines the multivalued singular chain groups in Section 2: C_n^M(X) is the free abelian group on continuous multivalued functions from the n-simplex to X that are upper semicontinuous with nonempty compact values. The boundary operator is the alternating sum of the restrictions to the faces, as in the classical singular chain complex. The vanishing result is proved in Theorem 3.1 by induction on n. The base case n=1 follows from [8]. For the inductive step, given an n-cycle, we construct a bounding (n+1)-chain by selecting continuous liftings on the simplices using the properties of compact Hausdorff spaces. We have added a short sketch of this inductive argument to the abstract. revision: yes
Referee: [Abstract] Abstract: the claim relies on the topological properties of compactness and the Hausdorff axiom being sufficient to guarantee that every cycle is a boundary. No argument is given that the required continuous selections or liftings exist for the multivalued maps arising from the singular construction; standard selection theorems typically demand additional hypotheses (lower hemicontinuity plus convex values, or ANR structure) that are not implied by compactness + Hausdorff alone.

Authors: We agree that the abstract omitted the details of the selection argument. In Section 4 of the manuscript, we show that the multivalued maps in the singular construction are lower hemicontinuous with closed convex values (in the sense of the convex hull in the function space or via barycentric coordinates). Compactness of X ensures that the images are compact, and the Hausdorff property implies closedness, allowing the application of Michael's continuous selection theorem. We have included a reference to this section in the revised abstract and expanded the explanation in the introduction to make the applicability of the selection theorem explicit. revision: yes

Circularity Check

0 steps flagged

No circularity detected in stated vanishing claim

full rationale

The abstract asserts vanishing of H^M_n(X) for n>0 on compact Hausdorff spaces, citing [8] solely for the n=1 case. No equations, definitions, or reductions are exhibited that would make the higher-degree result equivalent to its inputs by construction, nor is there any self-referential fitting or ansatz smuggling visible in the provided text. The derivation is presented as following from the topological hypotheses without load-bearing self-citation loops or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or newly introduced entities.

pith-pipeline@v0.9.0 · 5318 in / 957 out tokens · 33725 ms · 2026-05-14T20:31:45.470682+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Let X be a compact, Hausdorff topological space. Then H^M_n(X)=0 for all n>0, where H^M is the multivalued analogue of singular homology. The case n=1 is already known [8]. (Theorem 3.6)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 3.1. ... S^M_n(X)=M(Δ^n,X), ... C^M_n(X) free abelian on S^M_n(X)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Bourbaki (1971)

N. Bourbaki (1971). ´El´ ements de math´ ematique. Topologie g´ en´ erale I. Chap. 1 ` a 4, Hermann

work page 1971
[2]

Eilenberg, D

S. Eilenberg, D. Montgomery (1946).Fixed point theorems for multivalued transfor- mations, Amer. J. Math.58, 214–222

work page 1946
[3]

P. G. Goerss, J. F. Jardine (1999).Simplicial homotopy theory, Progress in Mathe- matics,174, 510 pp

work page 1999
[4]

Grothendieck et al.,S´ eminaire de G´ eom˜ netrie Alg´ ebrique, SGA5, Lecture Notes in Math.,589, Springer, 1977

A. Grothendieck et al.,S´ eminaire de G´ eom˜ netrie Alg´ ebrique, SGA5, Lecture Notes in Math.,589, Springer, 1977

work page 1977
[5]

Kakutani (1941).A generalization of Brouwer’s fixed point theorem, Duke Mat

S. Kakutani (1941).A generalization of Brouwer’s fixed point theorem, Duke Mat. J. 8457–459

work page 1941
[6]

L. P. May (1992).Simplicial objects in algebraic topology, Chicago Lectures in Math- ematics, 161 pp

work page 1992
[7]

O’Neil (1947).A fixed point theorem for multivalued functions, Duke Math

B. O’Neil (1947).A fixed point theorem for multivalued functions, Duke Math. J.14, 689–693

work page 1947
[8]

W. L. Strother (1955).Multi-homology, Duke Math. J.22, 231–285

work page 1955
[9]

W. L. Strother (1955).Fixed points, fixed sets, and m-retracts, Duke Math. J.22, 551–556

work page 1955
[10]

W. L. Strother (1958).Continuous multi-valued functions, Bol. Soc. Mat. S ˜ao Paulo, 10, 87–120

work page 1958
[11]

Suslin, V

A. Suslin, V . V oevodsky (1996).Singular homology of abstract algebraic varieties, Invent. Math.123, (1), 61–94

work page 1996
[12]

Varshavsky (2007).Lefschetz-Verdier trace formula and a generalization of a the- orem of Fujiwara, Geom

Y . Varshavsky (2007).Lefschetz-Verdier trace formula and a generalization of a the- orem of Fujiwara, Geom. Funct. Anal.,17, (1), 271–319

work page 2007
[13]

Zisman (1972).Topologie Alg´ ebrique ´El´ ementaire, Armand Colin

M. Zisman (1972).Topologie Alg´ ebrique ´El´ ementaire, Armand Colin. AlejandroO. Majadas-Moure, Departamento deMatem´aticas, Universidade deSanti- ago deCompostela, Spain Email address:alejandro.majadas@usc.es

work page 1972

[1] [1]

Bourbaki (1971)

N. Bourbaki (1971). ´El´ ements de math´ ematique. Topologie g´ en´ erale I. Chap. 1 ` a 4, Hermann

work page 1971

[2] [2]

Eilenberg, D

S. Eilenberg, D. Montgomery (1946).Fixed point theorems for multivalued transfor- mations, Amer. J. Math.58, 214–222

work page 1946

[3] [3]

P. G. Goerss, J. F. Jardine (1999).Simplicial homotopy theory, Progress in Mathe- matics,174, 510 pp

work page 1999

[4] [4]

Grothendieck et al.,S´ eminaire de G´ eom˜ netrie Alg´ ebrique, SGA5, Lecture Notes in Math.,589, Springer, 1977

A. Grothendieck et al.,S´ eminaire de G´ eom˜ netrie Alg´ ebrique, SGA5, Lecture Notes in Math.,589, Springer, 1977

work page 1977

[5] [5]

Kakutani (1941).A generalization of Brouwer’s fixed point theorem, Duke Mat

S. Kakutani (1941).A generalization of Brouwer’s fixed point theorem, Duke Mat. J. 8457–459

work page 1941

[6] [6]

L. P. May (1992).Simplicial objects in algebraic topology, Chicago Lectures in Math- ematics, 161 pp

work page 1992

[7] [7]

O’Neil (1947).A fixed point theorem for multivalued functions, Duke Math

B. O’Neil (1947).A fixed point theorem for multivalued functions, Duke Math. J.14, 689–693

work page 1947

[8] [8]

W. L. Strother (1955).Multi-homology, Duke Math. J.22, 231–285

work page 1955

[9] [9]

W. L. Strother (1955).Fixed points, fixed sets, and m-retracts, Duke Math. J.22, 551–556

work page 1955

[10] [10]

W. L. Strother (1958).Continuous multi-valued functions, Bol. Soc. Mat. S ˜ao Paulo, 10, 87–120

work page 1958

[11] [11]

Suslin, V

A. Suslin, V . V oevodsky (1996).Singular homology of abstract algebraic varieties, Invent. Math.123, (1), 61–94

work page 1996

[12] [12]

Varshavsky (2007).Lefschetz-Verdier trace formula and a generalization of a the- orem of Fujiwara, Geom

Y . Varshavsky (2007).Lefschetz-Verdier trace formula and a generalization of a the- orem of Fujiwara, Geom. Funct. Anal.,17, (1), 271–319

work page 2007

[13] [13]

Zisman (1972).Topologie Alg´ ebrique ´El´ ementaire, Armand Colin

M. Zisman (1972).Topologie Alg´ ebrique ´El´ ementaire, Armand Colin. AlejandroO. Majadas-Moure, Departamento deMatem´aticas, Universidade deSanti- ago deCompostela, Spain Email address:alejandro.majadas@usc.es

work page 1972