Pith. sign in

REVIEW 1 major objections 6 minor 60 references

On measure homology of mildly wild spaces

T0 review · 1 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The paper proves that for two broad classes of 'mildly wild' spaces, the canonical map from singular homology to measure homology is injective in degrees at least two.

desk verdict Solid paper with genuine new injectivity results for measure homology on mildly wild spaces; proofs are dense and one lemma needs a small patch but the main arguments hold up. read the letter →

arxiv 2506.03368 v1 pith:L3AKBTWO submitted 2025-06-03 math.AT math.GT

Thilo Kuessner , Janusz Przewocki , Andreas Zastrow This is my paper
classification math.ATmath.GT MSC 55N1055N35
keywords measurehomologysingularinjectivitymildlywildspacesconvergentY-spacesgeneralizeduniversalcoveringGromovnormboundedcohomology
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 establishes conditions under which the canonical map from singular homology to measure homology is injective for spaces that are not homotopy equivalent to CW-complexes. It proves two broad theorems: one for spaces with a contractible generalized universal covering and countable fundamental group, where any homology class killed by the map must have zero Gromov norm; and one for countable unions of finite CW-complexes with well-behaved contractible intersection components, where injectivity holds in degrees $k \ge 2$. These results cover the convergent arcs space, its higher-dimensional analogues, and glued negatively curved or hyperbolic surfaces. The point is to extend the known CW-complex coincidence result to a class of 'mildly wild' spaces, making measure-homology tools usable where singular homology alone is insufficient.

What carries the argument

The central object is the canonical homomorphism $\iota_*$ from singular homology $H_*(X;\mathbb{R})$ to measure homology $H_*(X)$, whose chain complex uses compactly determined signed measures of bounded variation on the space of singular simplices $\mathrm{map}(\Delta^k, X)$. The proofs are carried by two main devices: for Theorem 0.1, a Borel-measurable fundamental domain for the deck-transformation action on a contractible generalized universal covering, which makes measurable bounded cohomology isometrically isomorphic to bounded cohomology; and for Theorem 0.2, a barycentric-subdivision decomposition of measure chains across the finite CW layers, with Baire-category and non-accumulation arguments ensuring that each simplex lies in one layer or in a contractible intersection neighbourhood, followed by a coning and straightening construction that produces singular boundaries from measure boundaries.

What would settle it

Exhibit a first-countable $T_4$ space that is a countable union of finite CW-complexes with contractible intersection components, deformation-retract relatively-open neighbourhoods, and no accumulating intersection components, for which some $2$-cycle is a measure boundary but not a singular boundary; such a cycle would be a nonzero kernel element and disprove Theorem 0.2. A direct computation of the canonical map for the paper's own Example 3.4, where the deformation-retract neighbourhood condition fails, would test how sharp the hypothesis is.

Watch

Extended reading notes

Core claim

On the paper's own terms: for a mildly wild space $X$ — either a second-countable $T_1$ space with countable fundamental group admitting a contractible generalized universal covering, or a first-countable $T_4$ space that is a countable union of finite CW-complexes whose intersection components are contractible, have deformation-retract neighbourhoods, and do not accumulate — the canonical homomorphism $\iota_* : H_k(X; \mathbb{R}) \to H_k(X)$ is injective in degrees $k \ge 2$ (and in degree $0$ by prior work). The first theorem places the kernel inside the zero-norm subspace for the Gromov norm, so injectivity follows whenever that norm is a genuine norm; the second theorem avoids the norm condition and directly converts measure boundaries into singular boundaries. Applications include convergent $Y$-spaces for any finite CW-complex $Y$, such as the convergent arcs space, and countable gluings of negatively curved manifolds or hyperbolic surfaces along geodesics with a uniform strip-width condition.

Load-bearing premise

The load-bearing premise in the main structural theorem is that intersection components of the CW pieces are contractible, admit deformation-retract neighbourhoods that are relatively open, and do not accumulate; this is exactly the 'mild' regularity that separates the theorem from genuinely wild spaces such as the Hawaiian earring.

Editorial extensions

If this is right

  • For every convergent $Y$-space with $Y$ a finite CW-complex, $\iota_*$ is injective in degrees $k \ge 2$ (Corollary 3.15), so the convergent arcs space and its higher-dimensional analogues behave like CW-complexes in these degrees.
  • For convergent $Y$-spaces built from a closed negatively curved manifold, and for countable families of hyperbolic surfaces glued along geodesics with the stated strip-width condition, injectivity holds in degrees $k \ge 2$ (Corollary 2.13).
  • Under a countable deck group and a contractible generalized universal covering, the kernel of $\iota_*$ consists only of zero-Gromov-norm classes, so proving the Gromov norm is a norm becomes a sufficient route to injectivity.
  • The measurable bounded cohomology of $X$ is isometrically isomorphic to its bounded cohomology in this setting (Proposition 2.8), matching the CW-complex situation.
  • Measure homology and measurable cohomology can be computed from simplices with all vertices at a basepoint whenever $X$ is covered by finitely many Borel sets with compact contractible closures (Lemma 5.2 and Corollary 5.4).

Reading between the lines

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

  • If the announced degree-$1$ version of Theorem 0.2 succeeds, the convergent arcs space would have injectivity in every degree, making the mild-wild classes match CW-complexes completely rather than only in degrees $k \ge 2$.
  • The Gromov-norm containment in Theorem 0.1 suggests a quantitative strengthening: the kernel of $\iota_*$ is exactly the zero-norm subspace, so any future computation showing the Gromov norm is a norm on a mildly wild space would automatically settle injectivity and could feed into proportionality-principle arguments for non-manifold spaces.
  • A testable extension is to replace finite CW-complexes by finite simplicial complexes and ask whether metric convergence conditions weaker than uniform convergence in the definition of convergent $Y$-spaces still admit deformation-retract neighbourhoods; the paper's Example 3.10 indicates uniform convergence is essential.
  • The basepoint reduction of Section 5 could make pointed simplices the natural computational tool for measure homology of wild spaces, turning questions about uncountable fundamental groups into descriptive-set-theoretic obstructions such as the nonexistence of a Borel section in Theorem 5.7.
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

1 major / 6 minor

Summary. The paper studies the canonical map ι∗ from singular homology with real coefficients to measure homology (Milnor–Thurston homology) for spaces that do not necessarily have the homotopy type of a CW-complex. The main results are two injectivity theorems. Theorem 0.1 asserts that if X is T1, second countable, has countable fundamental group, and admits a contractible generalized universal covering, then the kernel of ι∗ is contained in the zero-norm subspace for the Gromov norm; injectivity follows whenever the Gromov norm is an actual norm. Theorem 0.2 asserts that if X is a first-countable T4-space which is a countable union of finite CW-complexes whose intersections satisfy the five conditions (i)–(v), then ι_k is injective for k ≥ 2. Applications include convergent Y-spaces (Corollary 3.15) and spaces obtained by gluing negatively curved manifolds or hyperbolic surfaces (Corollary 2.13). The final section develops a basepoint reduction for measure homology and measurable bounded cohomology and proves a non-existence result for measurable sections when the fundamental group is uncountable.

Significance. The results are substantial: they extend a basic comparison theorem, known for CW-complexes, to a class of genuinely non-CW spaces, with concrete applications to spaces relevant for simplicial volume and proportionality questions. The paper is careful about the boundary between 'mildly wild' and 'really wild' spaces; Examples 3.3, 3.4, 3.5, and 3.7 and Theorem 5.7 show that the hypotheses are not vacuous and that some of them are necessary. The measurable fundamental domain construction in Section 2.3 and the basepoint reduction in Section 5 are likely to be useful independently. The proof of Theorem 0.2, in particular Lemma 3.12 and Lemma 3.14, is intricate, and the authors are explicit about the k = 1 case remaining open. Overall, if the issues below are fixed, the paper would be a valuable contribution to the field.

major comments (1)
  1. [Section 2.3, Lemma 2.3] The proof of Lemma 2.3 applies the Lindelöf property to the family {U_x = p(\tilde U_{\tilde x})} and concludes that a countable subfamily covers X. However, the sets U_x need not be open, so the Lindelöf property does not apply to this family. Since Lemma 2.3 is used in the proof of Theorem 0.1, the proof of Theorem 0.1 is incomplete as written. A repair is available under the countable-π1 hypothesis of Theorem 0.1: one should first take a countable open cover of X\N by sets V_i with trivial π1-image, then use countability of the deck group to choose countably many components of p^{-1}(V_i) whose images still cover X\N, and then run the construction with these components. The lemma should either be stated with the additional hypothesis that the deck group is countable or be given a proof that works for uncountable deck groups.
minor comments (6)
  1. [Section 2.1] The same symbol C_k^b(X) is used for bounded cochains (Definition 1.4) and for measurable bounded cochains (Definition 2.1), and the notation H_b^*(X) is used for both resulting cohomology theories. This makes Proposition 2.8 and the surrounding text difficult to parse; distinct notation (for example, a calligraphic symbol for the measurable bounded cochains) would greatly improve readability.
  2. [Section 3.3, Lemma 3.14] The sentence introducing the contradiction argument in the proof of Lemma 3.14 appears to have a missing negation: as printed it says 'assume that this does not suffice, so that ... D_m only would contain simplices in ∪ M_i ∪ ∪ M_I', which is the desired conclusion. The intended assumption should be that for every m there is a simplex in D_m outside that union.
  3. [Section 5.4, Lemma 5.6] In the proof of Lemma 5.6(iv), the sentence 'str(K) is a finite set by (iii)' uses part (iii), which assumes that X is aspherical, but part (iv) is stated for an arbitrary countable CW-complex. Either add the asphericity hypothesis to (iv) or supply a separate argument for finiteness of str(K).
  4. [Section 4] In the completion of the proof of Corollary 2.13(b), the countability of π1(X) is not explicitly verified. It follows from the explicit countable construction of \tilde X in Construction 4.3, but this verification should be stated for completeness.
  5. [Section 3.4] In the proof of Theorem 0.2, the cancellation Σ_i v_i = 0 would be clearer if the authors stated explicitly that all filling chains v_i are produced by one fixed linear coning operator on the disjoint union of the contractible intersection components. The current phrase 'will automatically cancel' leaves the linearity step implicit.
  6. [Throughout] There are numerous typos and small grammatical slips (e.g., 'different different complex' after equation (0.5), 'a a class' before Corollary 2.13, and 'by by our Theorem' later in the introduction). A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from prior CW-complex results and new geometric/measure arguments, not from their own conclusions.

full rationale

The derivation chain is self-contained relative to prior external theorems; no step defines its output in terms of its input. Theorem 0.1 reduces injectivity to vanishing of the Gromov norm via Lemma 2.2, then proves the needed measurable bounded cohomology isomorphism through a measurable fundamental domain (Lemma 2.3), a measurable coning construction (Lemmas 2.5-2.7), and countability of the deck group. None of these steps assumes ker(ι_*) ⊂ N H. Theorem 0.2 starts from a singular cycle that is a measure boundary, decomposes the measure chain by barycentric subdivision (Lemma 3.14), and invokes the isomorphism ι_* : H_*(Y_i; R) → H_*(Y_i) only for the finite CW-complex pieces Y_i, citing [57] and [28]; this is a strictly weaker and independently established statement than the global injectivity being proved, so it is not circular. The Gromov-norm lower bounds in Sections 2.8 and 4 are obtained from curvature, straightening, and volume estimates, not fitted to the conclusion. The self-citations that appear ([19], [48], [57], [58]) supply definitions, prior theorems, and examples; none smuggles in the target injectivity as an unverified premise. The one in-scope technical concern is in Lemma 2.3: the sets U_x = p(UU~_x) need not be open, so the sentence 'Second-countable spaces have the Lindelöf property and hence there is a countable family of Ux that covers X' is not literally justified as written. That is a patchable proof gap, not a circular dependence, and it does not affect the score under the circularity rubric.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

Everything the proofs need beyond the paper's own arguments is in this list. Crucially, there are no fitted constants, no hidden normalizations, and no new postulated entities. The most contentful assumptions are the topological hypotheses in Theorems 0.1 and 0.2, which are stated honestly; the rest are standard background results from the cited literature (some authored by the same researchers, but not circularly used).

assumptions (6)
  • standard math Axiom of choice and existence of Borel sets/fundamental domains in the sense of Bourbaki
    Used in Lemma 1.8 to pick a representative from each deck-transformation orbit, in Example 3.1 to build a surjection π1(HA)→Q, and in Lemma 2.3 to construct a measurable fundamental domain. The paper explicitly notes the AC-dependence of the [49] counterexample.
  • domain assumption X has countable fundamental group and admits a generalized universal covering eX (Theorem 0.1 hypotheses)
    These hypotheses make the deck group G controllable and give the lifting structure used in Lemmas 1.8, 2.3 and Proposition 2.8; countability is essential for the measurable-section isomorphism.
  • domain assumption Theorem 0.2's intersection-component conditions (i)-(v): finite cell-subdivision, contractibility, deformation-retractible relatively open neighbourhoods, disjointness, and no accumulation
    These are the specific 'mildly wild' hypotheses that drive Lemma 3.12 (finite segmentation of simplices), the barycentric subdivision decomposition in Lemma 3.14, and the final cancellation argument. They are restrictive enough to exclude the Hawaiian earring.
  • standard math Measure homology satisfies the Eilenberg-Steenrod axioms, hence ι_* is an isomorphism for CW-complexes ([57],[28])
    Invoked in the proof of Theorem 0.2 to convert a measure-boundary in a finite CW-complex Y_i into a singular boundary, the key local injectivity step.
  • standard math Known theorems on generalized universal coverings: existence implies homotopically Hausdorff, path lifting, deck group G=π1, G acts freely/transitively on fibres ([19])
    Used throughout Section 2, including Lemma 2.3's proof that the countable non-slsc condition is automatic in the theorem's setting, and in Corollaries 2.13 and Section 4.
  • standard math Baire category theorem; metrizability/Polish space facts; Milnor's loop-space CW theorem; hyperbolic geometry/CAT(K) straightening facts
    Used in Lemma 3.12 (Baire theorem), Theorem 5.7 (Borel sets in Polish spaces), Lemma 2.11 (loop spaces), and Section 4 (straight simplices, tripods, volume bounds).

how reviews work

0 comments
Cite this review

Pith. "Pith review of On measure homology of mildly wild spaces." pith.science (2026). https://pith.science/paper/L3AKBTWO

@misc{pith2026250603368,
  author       = {Pith},
  title        = {Pith review of: On measure homology of mildly wild spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L3AKBTWO}},
  note         = {Machine review of arXiv:2506.03368}
}
read the original abstract

We prove injectivity of the canonical map from singular homology to measure homology for certain ``mildly wild" spaces, that is, certain spaces not having the homotopy type of a CW-complex, but having countable fundamental groups.

Figures

Figures reproduced from arXiv: 2506.03368 by the authors.

Figure 3
Figure 3. Fig.3.1]) one easily constructs hyperbolic surfaces with arbitrarily short in [PITH_FULL_IMAGE:figures/full_fig_p049_3.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages

  1. [1]

    Weighted combinatorial group theory and wild metric complexes

    W. A. Bogley, A. J. Sieradski: “Weighted combinatorial group theory and wild metric complexes”. In: “Groups-Korea 98” (edited by Y.G. Baik, D.L. Johnson, and A.C. Kim), de Gruyter, Berlin, New York, 2000, pages 53-80

    work page 2000

  2. [2]

    The word problem for some uncountable groups given by countable words

    O. Bogopolski and A. Zastrow: “The word problem for some uncountable groups given by countable words”, Topology Appl., Vol. 159 (2012), no. 3, 569–586

    work page 2012

  3. [3]

    Integration. II. Chapters 7-9

    N. Bourbaki: “Integration. II. Chapters 7-9.” Translated from the 1963 and 1969 French originals by Sterling K. Berberian. Elements of Mathematics (Berlin). Springer-Verlag, Berlin (2004)

    work page 2004

  4. [4]

    The uncountability of the Harmonic Archipelago Group

    J. Brazas: “The uncountability of the Harmonic Archipelago Group”, Blog- post in the blog on “Wild Topology”, maintained by J. Brazas (West Chester University, Pennsylvania, USA), posted by J. Brazas at 23 May, 2014, https://wildtopology.com/2014/05/23/the-uncountability-of-harmonic-archipelago- group/

    work page 2014

  5. [5]

    Metric Spaces of Non-Positive Curvature

    M. R. Bridson, A Haefliger: “Metric Spaces of Non-Positive Curvature”, Grundlehren der mathematischen Wissenschaften Vol. 319, Springer-Verlag, Berlin Heidelberg, 1999

    work page 1999

  6. [6]

    The simplicial volume of 3-manifolds with boundary

    M. Bucher, R. Frigerio, C. Pagliantini: “The simplicial volume of 3-manifolds with boundary.” J. Topol. 8, 457-475 (2015)

    work page 2015

  7. [7]

    Geometry and spectra of compact Riemann surfaces

    P. Buser’ “Geometry and spectra of compact Riemann surfaces.”, Progress in Math- ematics Vol. 106, Birkh¨ auser Boston, Inc., Boston, (Massachusetts, USA), 1992

    work page 1992

  8. [8]

    The combinatorial structure of the Hawaiian earring group

    J.W. Cannon and G.R. Conner: “The combinatorial structure of the Hawaiian earring group”, Topology Appl., Vol. 106 (2000), no. 3, 225-271. 68 THILO KUESSNER, JANUSZ PRZEWOCKI, AND ANDREAS ZASTROW

    work page 2000

Show all 60 references
  1. [9]

    On the fundamental groups of one-dimensional spaces

    J.W. Cannon, G.R. Conner: “On the fundamental groups of one-dimensional spaces”, Topology Appl., Vol. 153 (2006), no. 14, 2648-2672

    2006

  2. [10]

    Archipelago groups

    G. Conner, W. Hojka and M. Meilstrup: “Archipelago groups”, Proc. Amer. Math. Soc. 143 (2015), no. 11, 4973-4988

    2015

  3. [11]

    Covering maps for locally path- connected spaces

    N. Brodskiy, J. Dydak, B. Labuz and A. Mitra: “Covering maps for locally path- connected spaces”, Fund. Math. 218 (2012), no. 1, 13–46

    2012

  4. [12]

    Topology of metric complexes

    C. H. Dowker: “Topology of metric complexes.” Amer. J. Math. 74, 555–577 (1952)

    1952

  5. [13]

    Free sigma-products and non-commutatively slender groups

    K. Eda: “Free sigma-products and non-commutatively slender groups”, J. Algebra, Vol. 148 (1992), no. 1, 243-263

    1992

  6. [14]

    Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces

    K. Eda: “Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces”, J. Math. Soc. Japan, Vol. 63, no. 3 (2011), 769- 787

    2011

  7. [15]

    Singular homology groups of one-dimensional Peano continua

    K. Eda, “Singular homology groups of one-dimensional Peano continua”, Fund. Math., Vol. 232 (2016), no. 2, 99-115

    2016

  8. [16]

    The singular homology of the Hawaiian earring

    K. Eda and K. Kawamura: “The singular homology of the Hawaiian earring”, J. London Math. Soc. (2), Vol. 62 (2000), no. 1, 305-310

    2000

  9. [17]

    Singular homology theory

    S. Eilenberg: “Singular homology theory.” Ann. of Math. (2) 45, 407-447 (1944)

    1944

  10. [18]

    General Topology

    R. Engelking: “General Topology.” Pa´ nstwowe Wydawnictwo Naukowe, Warszawa (1977)

    1977

  11. [19]

    Generalized universal covering spaces and the shape group

    H. Fischer, A. Zastrow: “Generalized universal covering spaces and the shape group.” Fund. Math. 197, 167-196 (2007)

    2007

  12. [20]

    Proportionality principle for the Lipschitz simplicial volume

    F. Franceschini: “Proportionality principle for the Lipschitz simplicial volume.” Geom. Dedicata 182, 287-306 (2016)

    2016

  13. [21]

    (Bounded) continuous cohomology and Gromov’s proportionality prin- ciple

    R. Frigerio: “(Bounded) continuous cohomology and Gromov’s proportionality prin- ciple.” Manuscr. Math. 134, 435-474 (2011)

    2011

  14. [22]

    Cellular structures in topology

    R. Fritsch, R. Piccinini: “Cellular structures in topology.” Cambridge University Press, Cambridge (1990)

    1990

  15. [23]

    On arc-smooth continua

    J. B. Fugate, G. R. Gordh Jr., L. Lum: “On arc-smooth continua.” Top. Proc. 2, 645-656 (1977)

    1977

  16. [24]

    Arc-smooth continua

    J. B. Fugate, G. R. Gordh Jr., L. Lum: “Arc-smooth continua.” Trans. Amer. Math. Soc. 265, 545-561 (1981)

    1981

  17. [25]

    A generalization of the homology and homotopy suspension

    T. Ganea: “A generalization of the homology and homotopy suspension.” Comm. Math. Helv. 39, 295-322 (1964)

    1964

  18. [26]

    The fundamental group of two spaces with a common point

    H.B. Griffiths: “The fundamental group of two spaces with a common point”, Quart. J. Math., Oxford Ser. (2), Vol. 5, (1954). 175-190, and “The fundamental group of two spaces with a common point: a correction”, Quart. J. Math. Oxford Ser. (2), Vol. 6 (1955), 154-155 and “Infin...

    1954

  19. [27]

    Volume and bounded cohomology

    M. Gromov: “Volume and bounded cohomology.” Inst. Hautes `Etudes Sci. Publ. Math. 56, 5-99 (1983)

    1983

  20. [28]

    Measure homology

    S. K. Hansen: “Measure homology.” Math. Scand. 83, 205-219 (1998)

    1998

  21. [29]

    Algebraic Topology

    A. Hatcher: “Algebraic Topology.” Cambridge University Press, Cambridge (2002)

    2002

  22. [30]

    The harmonic archipelago as a universal locally free group

    W. Hojka: “The harmonic archipelago as a universal locally free group”, Journal of Algebra, Vol. 437 (2015), 44-51

    2015

  23. [31]

    The Gromov invariant of negatively curved manifolds

    H. Inoue, K. Yano: “The Gromov invariant of negatively curved manifolds.” Topology 21, 83-89 (1982)

    1982

  24. [32]

    Foundations of the theory of bounded cohomology

    N. Ivanov: “Foundations of the theory of bounded cohomology.” J. Sov. Math. 37, 1090-1115 (1987)

    1987

  25. [33]

    On the homology of the harmonic archipelago

    U. Karimov and D. Repovˇ s: “On the homology of the harmonic archipelago”, Cent. Eur. J. Math., Vol. 10 (2012), no. 3, 863-872

    2012

  26. [34]

    Classical descriptive set theory

    A. Kechris: “Classical descriptive set theory.” Springer Verlag, New York (1995)

    1995

  27. [35]

    Simplicial volume of compact manifolds with amenable bound- ary

    S. Kim, T. Kuessner: “Simplicial volume of compact manifolds with amenable bound- ary.” J. Topol. Anal. 7, 23-46 (2015)

    2015

  28. [36]

    Topological realisations of absolute Galois groups

    R. Kucharczyk, P. Scholze: “Topological realisations of absolute Galois groups.” in: J. W. Cogdell, G. Harder, S. Kudla, F. Shahidi (eds.): “Cohomology of arithmetic groups.” Springer Proceedings in Mathematics and Statistics 245, Springer, 201-288 (2018). ON MEASURE HOMOLOGY ...

    2018

  29. [37]

    Multicomplexes, bounded cohomology and additivity of simplicial vol- ume

    T. Kuessner: “Multicomplexes, bounded cohomology and additivity of simplicial vol- ume.” B. Korean Math. Soc. 52, 1855-1899 (2015)

    2015

  30. [38]

    Measure homology and singular homology are isometrically isomorphic

    C. L¨ oh: “Measure homology and singular homology are isometrically isomorphic.” Math. Z. 253, 197-218 (2006)

    2006

  31. [39]

    Degree theorems and Lipschitz simplicial volume for nonpositively curved manifolds of finite volume

    C. L¨ oh, R. Sauer: “Degree theorems and Lipschitz simplicial volume for nonpositively curved manifolds of finite volume.” J. Topol. 2, 193-225 (2009)

    2009

  32. [40]

    The topology of CW-complexes

    A. Lundell, S. Weingram: “The topology of CW-complexes.” Van Nostrand, New York (1969)

    1969

  33. [41]

    Simplicial objects in algebraic topology

    J. P. May: “Simplicial objects in algebraic topology.” Van Nostrand Mathemati- cal Studies No.11, D. Van Nostrand Co. Inc., Princeton, N.J.-Toronto, Ont.-London (1967)

    1967

  34. [42]

    A van Kampen theorem for weak joins

    J.W. Morgan, I. Morrison: “A van Kampen theorem for weak joins.” Proc. London Math. Soc. (3), Vol. 53 (1986), no. 3, 562-576

    1986

  35. [43]

    Continuous bounded cohomology of locally compact groups

    N. Monod: “Continuous bounded cohomology of locally compact groups.” Lecture Notes in Mathematics, 1758. Springer-Verlag, Berlin (2001)

    2001

  36. [44]

    Geometry and Spectrum of typical hyperbolic surfaces

    L. Monk: “Geometry and Spectrum of typical hyperbolic surfaces.” PhD-Thesis, Strasbourg University, 2021,

    2021

  37. [45]

    Elementary differential topology

    J. R. Munkres: “Elementary differential topology”, Annals of Mathematics Studies No.54 (rev. ed., 2-nd printing), Princeton University Press, Princeton (New Jersey, USA, 1968)

    1968

  38. [46]

    Atomic properties of the Hawaiian earring group for HNN-extensions

    J. Nakamura: “Atomic properties of the Hawaiian earring group for HNN-extensions”, Comm. Algebra, Vol. 43 (2015), no. 10, 4138-4147

    2015

  39. [47]

    Milnor-Thurston homology groups of the Warsaw Circle

    J. Przewocki: “Milnor-Thurston homology groups of the Warsaw Circle.” Topology Appl. 160, no. 13, 1732-1741 (2013)

    2013

  40. [48]

    Milnor-Thurston homology of some wild topological spaces

    J. Przewocki: “Milnor-Thurston homology of some wild topological spaces.” PhD- dissertation, Polish Academy of Sciences (2014)

    2014

  41. [49]

    On the coincidence of zeroth Milnor-Thurston homology with singular homology

    J. Przewocki, A. Zastrow: “On the coincidence of zeroth Milnor-Thurston homology with singular homology.” Fund. Math. 243, 109-122 (2018)

    2018

  42. [50]

    Omega-groups

    A. J. Sieradski: “Omega-groups”, unpublished notes, University of Oregon, Eugene (Oregon, USA), 1997/1998

    1997

  43. [51]

    A course on Borel sets

    S. Srivastava: “A course on Borel sets.” Springer Verlag, New York (1998)

    1998

  44. [52]

    Piecewise straightening and Lipschitz simplicial volume

    K. Strza lkowski: “Piecewise straightening and Lipschitz simplicial volume.” J. Topol. Anal. 9, 167-193 (2017)

    2017

  45. [53]

    W. P. Thurston: “Collected works of William P. Thurston with commentary: IV. The geometry and topology of three-manifolds: with a preface by Steven P. Kerckhoff. Edited by Benson Farb, David Gabai and Steven P. Kerckhoff.” Providence, RI: American Mathematical Society (2022)

    2022

  46. [54]

    Big free groups are almost free

    T. Tlas: “Big free groups are almost free”, Internat. J. Algebra Comput., Vol. 25 (2015), no. 5, 855-864

    2015

  47. [55]

    A homotopically Hausdorff space which does not admit a generalized universal covering space

    ˇZ. Virk and A. Zastrow: “A homotopically Hausdorff space which does not admit a generalized universal covering space”, Topology Appl., Vol. 160 (2013), no. 4, 656-666

    2013

  48. [56]

    Local connectivity of mapping spaces

    H. Wada: “Local connectivity of mapping spaces.” Duke Math. J. 22, 419-425 (1955)

    1955

  49. [57]

    On the (non)-coincidence of Milnor-Thurston homology theory with singular homology theory

    A. Zastrow: “On the (non)-coincidence of Milnor-Thurston homology theory with singular homology theory.” Pacific J. Math. 186, 369-396 (1998)

    1998

  50. [58]

    Zastrow, in preparation

    A. Zastrow, in preparation

  51. [59]

    Construction of an infinitely generated group that is not a free product of surface groups and abelian groups, but which acts freely on an R-tree

    A. Zastrow, “Construction of an infinitely generated group that is not a free product of surface groups and abelian groups, but which acts freely on an R-tree”, Proc. Roy. Soc. Edinburgh Sect. A, Vol. 128 (1998), no. 2, 433-445

    1998

  52. [60]

    The non-abelian Specker-group is free

    A. Zastrow, “The non-abelian Specker-group is free”, J. Algebra, Vol. 229 (2000), no. 1, 55-85. 70 THILO KUESSNER, JANUSZ PRZEWOCKI, AND ANDREAS ZASTROW Mathematisch-Geographische F akult¨at, Katholische Universit¨at Eichst¨att- Ingolstadt, Ostenstraße 28, 85072 Eichst ¨att, G...

    2000

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.