Celebratio Mathematica — Neumann — Complete Bibliography

[1] W. D. Neumann: “3-dimensional \( G \)-manifolds with 2-dimensional orbits,” pp. 220–222 in Proceedings of a conference on transformation groups (New Orleans, LA, 8 May–2 June 1967). Edited by P. S. Mostert. Springer (New York), 1968. MR 245043 Zbl 0177.52101 incollection

[2] W. D. Neumann: “On cardinalities of free algebras and ranks of operations,” Arch. Math. (Basel) 20 (1969), pp. 132–133. MR 248068 Zbl 0177.02601 article

[3] W. D. Neumann: “Representing varieties of algebras by algebras,” J. Aust. Math. Soc. 11 : 1 (1970), pp. 1–8. Dedicated to my father on his 60th birthday. MR 277465 Zbl 0199.32702 article

[4] W. D. Neumann: “On the quasivariety of convex subsets of affine spaces,” Arch. Math. (Basel) 21 (1970), pp. 11–16. MR 262911 Zbl 0194.01502 article

[5] W. D. Neumann: \( S^1 \)-actions and the \( \alpha \)-invariant of their involutions. Bonn Mathematical Texts 44. Universität Bonn, Mathematisches Institut, 1970. MR 317346 Zbl 0219.57030 book

[6] F. Hirzebruch, W. D. Neumann, and S. S. Koh: Differentiable manifolds and quadratic forms. Lecture Notes in Pure and Applied Mathematics 4. Marcel Dekker (New York), 1971. With an appendix by W. Scharlau. MR 341499 Zbl 0226.57001 book

F. Hirzebruch	Related	Volume
Winfried Scharlau	Related
Sebastian Su Koh	Related

[7] W. D. Neumann: “Fibering over the circle within a bordism class,” Math. Ann. 192 (1971), pp. 191–192. MR 287554 Zbl 0211.55602 article

[8] U. Karras, M. Kreck, W. D. Neumann, and E. Ossa: Cutting and pasting of manifolds; \( \mathrm{SK} \)-groups. Mathematics Lecture Series 1. Publish or Perish (Boston), 1973. MR 362360 Zbl 0258.57010 book

Matthias Kreck	Related
Erich Ossa	Related
Ulrich Karras	Related

[9] W. D. Neumann: “On Malcev conditions,” J. Aust. Math. Soc. 17 : 3 (1974), pp. 376–384. Dedicated to my mother, in loving memory. MR 371781 Zbl 0294.08004 article

This note gives a way of looking at Malcev conditions for varieties as ideals in a certain lattice. Though this viewpoint (so far) yields no new results, we feel it puts Walter Taylor’s results [1975] characterizing “Malcev definable classes of varieties” into a clearer perspective and is therefore worth mentioning.

We construct in a simple way a complete lattice \( L \) of equivalence classes of varieties containing a countable sublattice \( M \) (represented by the finitely presented varieties) such that:

equivalent varieties are indistinguishable by Malcev conditions, that is, they either all satisfy a given Malcev condition or all do not;
the classes of varieties defined by strong Malcev conditions are just (modulo the equivalence relation) the principal ideals in \( L \) generated by single elements of \( M \);
the classes of varieties defined by ordinary Malcev conditions (respectively, weak Malcev conditions) are the ideals in \( L \) generated by subsets of \( M \) (respectively, countable intersections of such ideals). Walter Taylor’s closure theorems follow easily.

Our lattice \( L \) is actually defined on a class, rather than a set. This can be avoided by a cardinality restriction on the number of defining operations that a variety may have; \( L \) will then be complete only up to this cardinality.

[10] W. D. Neumann: “Cyclic suspension of knots and periodicity of signature for singularities,” Bull. Am. Math. Soc. 80 : 5 (September 1974), pp. 977–981. MR 358797 Zbl 0292.57013 article

[11] U. Hirsch and W. D. Neumann: “On cyclic branched coverings of spheres,” Math. Ann. 215 (October 1975), pp. 289–291. MR 375321 Zbl 0289.57003 article

In the \( PL \) category every orientable closed \( n \)-manifold \( M^n \) can be exhibited as a branched covering of the \( n \)-sphere \( S^n \) [Alexander 1920]. If \( M^n \) is given one can therefore ask for the minimal integer \( d \geq 1 \) for which there exists such a map whose degree is equal to \( d \). If \( n = 2 \) then always \( d\leq 2 \). If \( n = 3 \) then \( d \leq 3 \), ([Hilden 1974; Hirsch 1974, §7.3; Montesinos 1974]). In general these maps are irregular.

In this note we are concerned however with a class of regular branched coverings of \( S^n \). With reference to papers by Fox [1972], Montesinos [1975], and Birman and Hilden [1973] it is shown that there is a big class of orientable closed manifolds which cannot be realized as cyclic, in particular as 2-fold branched coverings of the sphere.

[12] W. D. Neumann: “Manifold cutting and pasting groups,” Topology 14 : 3 (August 1975), pp. 237–244. MR 380837 Zbl 0311.57007 article

[13] M. Brown and W. D. Neumann: “Proof of the Poincaré–Birkhoff fixed point theorem,” Mich. Math. J. 24 : 1 (1977), pp. 21–31. MR 448339 Zbl 0402.55001 article

The Poincaré–Birkhoff fixed point theorem (also called Poincaré’s last geometric theorem) asserts the existence of at least two fixed points for a so-called area-preserving twist homeomorphism of the annulus. It was formulated as a conjecture and proved in special cases by Poincaré [1912], shortly before his death. In [1913] George Birkhoff published a proof which, though correct for one fixed point, overlooked the passibility that this fixed point might have index 0 in deducing the existence of a second fixed point. This error was corrected in his paper [1926], in which a generalization of the theorem in question is proven, with “area-preserving” replaced by a purely topological condition and “homeomorphism” replaced by a more general situation. However, some mathematicians have claimed that this proof too is incorrect, and the last few years have seen some extensive efforts to try to find a correct proof for the second fixed point.

We present here an elementary proof for two fixed points which is a simple modification of Birkhoff’s well known original proof for one fixed point. Our modification to get the second fixed point is essentially the same modification that Birkhoff sketches in the 1926 proof of his topological version to get from one fixed point to two.

This paper is therefore in a sense an expository paper, and to make the proof as transparent as possible we shall restrict to the simplest situation — a twist homeomorphism of the annulus which is just a rotation by a fixed angle on each boundary circle. As we point out in a final section, the proof goes through almost word for word without this restriction. It also extends to more general measures than the standard Lebesgue measure on the annulus.

Since our proof is so chase to Birkhoff’s proof, which has met with some skepticism, we have felt it advisable to give somewhat more detail than would otherwise be necessary. This is also in keeping with the view of this paper as an expository one.

[14] W. D. Neumann: “Brieskorn complete intersections and automorphic forms,” Invent. Math. 42 : 1 (1977), pp. 285–293. Gratefully dedicated to my teacher F. E. Hirzebruch. MR 463493 Zbl 0366.32015 article

[15] L. H. Kauffman and W. D. Neumann: “Products of knots, branched fibrations and sums of singularities,” Topology 16 : 4 (1977), pp. 369–393. MR 488073 Zbl 0447.57012 article

[16] W. D. Neumann: Equivariant Witt rings. Bonner Mathematische Schriften 100. Universität Bonn, Mathematisches Institut, 1977. MR 494248 Zbl 0368.10020 book

[17] W. D. Neumann: “Generalizations of the Poincaré Birkhoff fixed point theorem,” Bull. Aust. Math. Soc. 17 : 3 (1977), pp. 375–389. MR 584597 Zbl 0372.54041 article

[18] W. D. Neumann: “Mal’cev conditions, spectra and Kronecker product,” J. Aust. Math. Soc. Ser. A 25 : 1 (1978), pp. 103–117. Errata for this article were published in J. Aust. Math. Soc. 28:4 (1970). MR 480271 Zbl 0387.08004 article

[19] W. D. Neumann: “Multiplicativity of signature,” J. Pure Appl. Algebra 13 : 1 (1978), pp. 19–31. Dedicated to the memory of George Cooke. MR 508726 Zbl 0404.55008 article

[20] W. D. Neumann and S. H. Weintraub: “Four-manifolds constructed via plumbing,” Math. Ann. 238 : 1 (1978), pp. 71–78. MR 510309 Zbl 0372.57004 article

The following procedure seemed to present a way of constructing interesting 4-manifolds: plumb disc bundles with even euler numbers over 2-spheres together in such a way that the boundary of the resulting 4-manifold \( M_0^4 \) is simply connected, hence \( S^3 \) by von Randow [1962], Scharf [1975], and Montesinos [1975]. Then \( M = M_0^4 \cup D^4 \) is a 4-manifold with even intersection form.

We shall show that the 4-manifolds obtained this way are diffeomorphic to a connected sum of copies of \( S^2\times S^2 \). More generally:

If \( M^4 = M_0^4 \cup D^4 \), where \( M_0^4 \) is a 4-manifold obtained by plumbing with \( \partial M_0 = S^3 \), then either \[ M = (S^2\times S^2)\mathbin{\#}\cdots\mathbin{\#} (S^2\times S^2) \] or \[ M = CP^2 \mathbin{\#}\cdots\mathbin{\#} CP^2 \mathbin{\#}-CP^2\mathbin{\#}\cdots\mathbin{\#}-CP^2 ,\] according as \( M \) has even or odd intersection form.

[21] W. D. Neumann and F. Raymond: “Seifert manifolds, plumbing, \( \mu \)-invariant and orientation reversing maps,” pp. 163–196 in Algebraic and geometric topology: Proceedings of a symposium held at Santa Barbara in honor of Raymond L. Wilder (Santa Barbara, CA, 25–29 July 1977). Edited by K. C. Millett. Lecture Notes in Mathematics 664. Springer, 1978. MR 518415 Zbl 0401.57018 incollection

Kenneth C. Millett	Related
Raymond Louis Wilder	Related
Frank Albert Raymond	Related

[22] W. D. Neumann: “Homotopy invariance of Atiyah invariants,” pp. 181–188 in Algebraic and geometric topology (Stanford, CA, 2–21 August 1976), part 2. Edited by R. J. Milgram. Proceedings of Symposia in Pure Mathematics 32. American Mathematical Society (Providence, RI), 1978. MR 520534 Zbl 0399.55013 incollection

[23] W. D. Neumann: “Signature related invariants of manifolds, I: Monodromy and \( \gamma \)-invariants,” Topology 18 : 2 (1979), pp. 147–172. MR 544156 Zbl 0416.57013 article

[24] W. D. Neumann: “Normal subgroups with infinite cyclic quotient,” Math. Sci. 4 : 2 (1979), pp. 143–148. MR 562003 Zbl 0414.20030 article

[25] W. D. Neumann: “Errata: ‘Mal’cev conditions, spectra and Kronecker product’,” J. Austral. Math. Soc. Ser. A 28 : 4 (1979), pp. 510. Errata for an article published in J. Aust. Math. Soc. 25:1 (1978). MR 562882 Zbl 0419.08009 article

[26] W. D. Neumann: “An invariant of plumbed homology spheres,” pp. 125–144 in Topology symposium (Siegen, Germany, 14–19 June 1979). Edited by U. Koschorke and W. D. Neumann. Lecture Notes in Mathematics 788. Springer (Berlin), 1980. MR 585657 Zbl 0436.57002 incollection

[27] W. D. Neumann: “A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves,” Trans. Am. Math. Soc. 268 : 2 (1981), pp. 299–344. MR 632532 Zbl 0546.57002 article

[28] D. Eisenbud, U. Hirsch, and W. Neumann: “Transverse foliations of Seifert bundles and self-homeomorphism of the circle,” Comment. Math. Helv. 56 : 4 (1981), pp. 638–660. MR 656217 Zbl 0516.57015 article

David Eisenbud	Related
Ulrich Hirsch	Related

[29] W. D. Neumann: “Abelian covers of quasihomogeneous surface singularities,” pp. 233–243 in Singularities (Arcata, CA, 20 July–7 August 1981), part 2. Edited by P. Orlik. Proceedings of Symposia in Pure Mathematics 40. American Mathematical Society (Providence, RI), 1983. MR 713252 Zbl 0519.32010 incollection

[30] W. D. Neumann: “Geometry of quasihomogeneous surface singularities,” pp. 245–258 in Singularities (Arcata, CA, 20 July–7 August 1981), part 2. Edited by P. Orlik. Proceedings of Symposia in Pure Mathematics 40. American Mathematical Society (Providence, RI), 1983. MR 713253 Zbl 0519.32011 incollection

[31] W. D. Neumann: “Invariants of plane curve singularities,” pp. 223–232 in Nœuds, tresses et singularités [Knots, braids and singularities] (Plans-sur-Bex, Switzerland, 27 March–2 April 1982). Edited by C. Weber. Monographie de l’Enseignement Mathématique 31. University of Geneva, 1983. MR 728588 Zbl 0586.14023 incollection

[32] M. Jankins and W. D. Neumann: Lectures on Seifert manifolds. Brandeis Lecture Notes 2. Brandeis University (Waltham, MA), 1983. MR 741334 book

[33] M. Jankins and W. D. Neumann: “Rotation numbers of products of circle homeomorphisms,” Math. Ann. 271 : 3 (1985), pp. 381–400. MR 787188 Zbl 0543.57019 article

[34] M. Jankins and W. Neumann: “Homomorphisms of Fuchsian groups to \( \textrm{PSL}(2,\textrm{R}) \),” Comment. Math. Helv. 60 : 3 (1985), pp. 480–495. MR 814153 Zbl 0598.57007 article

[35] W. D. Neumann and D. Zagier: “Volumes of hyperbolic three-manifolds,” Topology 24 : 3 (1985), pp. 307–332. MR 815482 Zbl 0589.57015 article

By “hyperbolic 3-manifold” we will mean an orientable complete hyperbolic 3-manifold \( M \) of finite volume. By Mostow rigidity the volume of \( M \) is a topological invariant, indeed a homotopy invariant, of the manifold \( M \). There is in fact a purely topological definition of this invariant, due to Gromov. The set of all possible volumes of hyperbolic 3-manifolds is known to be a well-ordered subset of the real numbers and is of considerable interest (for number theoretic aspects see, for instance, [Borel 1981; Zagier 1986]) but remarkably little is known about it: the smallest element is not known even approximately, and it is not known whether any element of this set is rational or whether any element is irrational. For more details see Thurston’s Notes [1977]. In this paper we prove a result which, among other things, gives some metric or analytic information about the set of hyperbolic volumes.

[36] D. Eisenbud and W. Neumann: Three-dimensional link theory and invariants of plane curve singularities. Annals of Mathematics Studies 110. Princeton University Press, 1985. MR 817982 Zbl 0628.57002 book

[37] W. D. Neumann and D. Zagier: “A note on an invariant of Fintushel and Stern,” pp. 241–244 in Geometry and topology (College Park, MD, 1983–1984). Edited by J. C. Alexander and J. L. Harer. Lecture Notes in Mathematics 1167. Springer, 1985. MR 827273 Zbl 0589.57016 incollection

James Crew Alexander	Related
Don Bernard Zagier	Related
John L. Harer	Related

[38] W. D. Neumann: “Splicing algebraic links,” pp. 349–361 in Complex analytic singularities (Ibaraki, Japan, 16–20 July 1984). Edited by T. Suwa and P. Wagreich. Advanced Studies in Pure Mathematics 8. North-Holland (Amsterdam), 1987. MR 894301 Zbl 0652.32011 incollection

In this paper we give an introduction to the terminology of splicing (see [Eisenbud and Neumann 1985]) and then describe how to compute a normal form representation of the real monodromy and Seifert form for the link of a plane curve singularity from this point of view (it was done via a resolution diagram for the singularity in [Neumann 1983]). It has been conjectured that this might be a complete invariant for the topology of an isolated complex hypersurface singularity in any dimension; the originator now denies responsibility and will remain unnamed, but the conjecture is still unresolved. Many of the required invariants are computed in [Eisenbud and Neumann 1985] and we just review these computations. The first four sections and Theorem 5.1 are survey and review; the main new result is the computation of the equivariant signatures of the monodromy via splicing in Theorem 5.3. This computation applies also to general graph links.

[39] F. Ehlers, W. D. Neumann, and J. Scherk: “Links of surface singularities and CR space forms,” Comment. Math. Helv. 62 : 2 (1987), pp. 240–264. In memory of Peter Scherk. MR 896096 Zbl 0626.32032 article

Peter Scherk	Related
John Scherk	Related
Fritz Dirk Ehlers	Related

[40] W. Neumann and L. Rudolph: “Unfoldings in knot theory,” Math. Ann. 278 : 1–4 (1987), pp. 409–439. Dedicated to Friedrich Hirzebruch. A corrigendum to this article was published in Math. Ann. 282:2 (1989). MR 909235 Zbl 0675.57010 article

F. Hirzebruch	Related	Volume
Lee Norman Rudolph	Related

[41] R. Bieri, W. D. Neumann, and R. Strebel: “A geometric invariant of discrete groups,” Invent. Math. 90 : 3 (1987), pp. 451–477. MR 914846 Zbl 0642.57002 article

Ralph Strebel	Related
Robert Bieri	Related

[42] W. Neumann and L. Rudolph: “Corrigendum: ‘Unfoldings in knot theory’,” Math. Ann. 282 : 2 (1988), pp. 349–351. Corrigendum to article published in Math. Ann. 278:1–4 (1987). MR 963022 Zbl 0675.57011 article

[43] W. D. Neumann and L. Rudolph: “The enhanced Milnor number in higher dimensions,” pp. 109–121 in Differential topology (Siegen, Germany, 27 July–1 August 1987). Lecture Notes in Mathematics 1350. Springer (Berlin), 1988. MR 979336 Zbl 0655.57015 incollection

[44] W. D. Neumann: “On bilinear forms represented by trees,” Bull. Aust. Math. Soc. 40 : 2 (1989), pp. 303–321. MR 1012837 Zbl 0686.05018 article

[45] W. D. Neumann: “Complex algebraic plane curves via their links at infinity,” Invent. Math. 98 : 3 (1989), pp. 445–489. MR 1022302 Zbl 0734.57011 article

Considering that the study of plane cuves has an over 2000 year history and is the seed from which modern algebraic geometry grew, surprisingly little is known about the topology of affine algebraic plane curves. We topologically classify “regular” algebraic plane curves in complex affine 2-space using “splice diagrams:” certain decorated trees that code Puiseux data at infinity. (The regularity condition — that the curve be a “typical” fiber of its defining polynomial — can conjecturally be avoided.) We also show that the splice diagram determines such algebraic information as the minimal degree of the curve, even in the irregular case. Among other things, this enables algebraic classification of regular algebraic plane curves with given topology.

[46] W. D. Neumann: “On the topology of curves in complex surfaces,” pp. 117–133 in Topological methods in algebraic transformation groups (New Brunswick, NJ, 4–8 April 1988). Edited by H. Kraft, T. Petrie, and G. W. Schwarz. Progress in Mathematics 80. Birkhäuser (Boston), 1989. MR 1040860 Zbl 0715.14025 incollection

We describe the use of the link at infinity as a tool to classify complex affine plane curves. This is an exposition of results of [Neumann 1989]. We give an overview of these results in §1, describe them precisely in §2, and give more examples in §4. §3 describes one of the topological ingredients: a substitute for the “Milnor fibration at infinity” when the latter does not exist.

The basic philosophy is that a three-manifold has a quite transparent structure given by its “toral decomposition,” and that for a three-manifold link of a complex analytic object this structure is induced from the structure in the complex object, and thus tells one a lot about it.

The same philosophy is useful also in other situations, and in §5 we make some remarks on links of divisors in surfaces, which we apply to show how the link at infinity of an affine homology plane determines the compactification divisor. This part of the paper is independent from the foregoing; it expands on a short digression in the conference talk.

Ted E. Petrie	Related
Gerald Walter Schwarz	Related

[47] W. Neumann and J. Wahl: “Casson invariant of links of singularities,” Comment. Math. Helv. 65 : 1 (1990), pp. 58–78. MR 1036128 Zbl 0704.57007 article

[48] W. D. Neumann and L. Rudolph: “Difference index of vectorfields and the enhanced Milnor number,” Topology 29 : 1 (1990), pp. 83–100. MR 1046626 Zbl 0760.57003 article

[49] W. D. Neumann: “On intersections of finitely generated subgroups of free groups,” pp. 161–170 in Groups: Canberra 1989 (Canberra, 25–29 September 1989). Edited by L. G. Kovacs. Lecture Notes in Mathematics 1456. Springer (Berlin), 1990. MR 1092229 Zbl 0722.20016 incollection

[50] W. D. Neumann and A. W. Reid: “Amalgamation and the invariant trace field of a Kleinian group,” Math. Proc. Cambridge Philos. Soc. 109 : 3 (1991), pp. 509–515. MR 1094749 Zbl 0728.57009 article

[51] R. Meyerhoff and W. D. Neumann: “An asymptotic formula for the eta invariants of hyperbolic 3-manifolds,” Comment. Math. Helv. 67 : 1 (1992), pp. 28–46. MR 1144612 Zbl 0791.57009 article

Let \( M \) be an oriented complete finite-volume hyperbolic 3-manifold with one cusp. Suppose that an oriented basis \( \mathbf{l} \), \( \mathbf{m} \), for the first homology at the cusp has been chosen, so that we can speak of \( M(p,q) \), the result of \( (p,q) \)-Dehn filling the cusp — that is, we replace the cusp by a solid torus in which the class \( p\mathbf{m} + q\mathbf{l} \) is null-homologous. Thurston’s hyperbolic Dehn surgery theorem [Thurston 1978; Neumann and Zagier 1985] tells us that \( M(p,q) \) has a hyperbolic structure for \( p^2 + q^2 \) sufficiently large. In [1985] T. Yoshida proves a formula for the eta invariant \( \eta(M(p,q)) \) in terms of Thurston’s analytic Dehn surgery parameter \( u(p,q) \) and additional structure on \( M \) (various frame fields). For \( M \) equal to the figure-eight knot complement he gives a simpler and more explicit version of the formula which does not invoke the extra structure. The purpose of this note is to show that a formula of this simpler type can be derived in general from Yoshida’s result.

[52] W. D. Neumann: “Asynchronous combings of groups,” Internat. J. Algebra Comput. 2 : 2 (1992), pp. 179–185. MR 1176383 Zbl 0777.20013 article

[53] W. D. Neumann: “The fixed group of an automorphism of a word hyperbolic group is rational,” Invent. Math. 110 : 1 (1992), pp. 147–150. MR 1181820 Zbl 0793.20033 article

[54] Topology ’90: Papers from the research semester in low-dimensional topology held at Ohio State University (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematical Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184397 Zbl 0747.00024 book

Boris Nikolaevich Apanasov	Related
Laurent C. Siebenmann	Related
Alan William Reid	Related

[55] W. D. Neumann: “Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic 3-manifolds,” pp. 243–271 in Topology ’90: Papers from the research semester in low-dimensional topology held at Ohio State University (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematics Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184415 Zbl 0768.57006 incollection

Boris Nikolaevich Apanasov	Related
Laurent C. Siebenmann	Related
Alan William Reid	Related

@incollection {key1184415m,
	AUTHOR = {Neumann, Walter D.},
	TITLE = {Combinatorics of triangulations and
		 the {C}hern--{S}imons invariant for
		 hyperbolic 3-manifolds},
	BOOKTITLE = {Topology '90: {P}apers from the research
		 semester in low-dimensional topology
		 held at {O}hio {S}tate {U}niversity},
	EDITOR = {Apanasov, Boris and Neumann, Walter
		 D. and Reid, Alan W. and Siebenmann,
		 Laurent},
	SERIES = {Ohio State University Mathematics Research
		 Institute Publications},
	NUMBER = {1},
	PUBLISHER = {de Gruyter},
	ADDRESS = {Berlin},
	YEAR = {1992},
	PAGES = {243--271},
	URL = {https://www.degruyter.com/document/doi/10.1515/9783110857726/pdf#page=255},
	NOTE = {(Columbus, OH, February--June 1990).
		 MR:1184415. Zbl:0768.57006.},
	ISSN = {0942-0363},
	ISBN = {9783110125986},
}

[56] W. D. Neumann and A. W. Reid: “Arithmetic of hyperbolic manifolds,” pp. 273–310 in Topology ’90: Papers from the research semester in low-dimensional topology held at Ohio State University (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematics Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184416 Zbl 0777.57007 incollection

Boris Nikolaevich Apanasov	Related
Laurent C. Siebenmann	Related
Alan William Reid	Related

[57] W. D. Neumann and A. W. Reid: “Notes on Adams’ small volume orbifolds,” pp. 311–314 in Topology ’90: Papers from the research semester in low-dimensional topology held at Ohio State University (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematics Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184417 Zbl 0773.57009 incollection

In [1992] Colin Adams shows there we just six non-compact orientable hyperbolic 3-orbifolds of volume less than \( v_0/4 \), where \( v_0 = 1.01494146\dots \) is the volume of a regular ideal tetrahedron in \( \mathbb{H}^3 \). We give various descriptions of them and their fundamental groups. In particular, we will see that they are all arithmetic, and we give explicit commensurabilities between some of them. Their fundamental groups can be described as symmetry groups of tesselations, so we will introduce terminology for this. Many of them am also “tetrahedral groups,” so we introduce notation for these too.

These orbifolds show that “cusp density” is not a commensurability invariant of one-cusped orbifolds, answering a question that has been asked by several people. As we discuss in Sect. 3, cusp density is a commensurability invariant for non-arithmetic one-cusped orbifolds.

Boris Nikolaevich Apanasov	Related
Laurent C. Siebenmann	Related
Alan William Reid	Related

[58] W. D. Neumann and M. Shapiro: “Equivalent automatic structures and their boundaries,” Int. J. Algebra Comput. 2 : 4 (1992), pp. 443–469. MR 1189673 Zbl 0767.20013 article

[59] W. D. Neumann and A. W. Reid: “Rigidity of cusps in deformations of hyperbolic 3-orbifolds,” Math. Ann. 295 : 2 (1993), pp. 223–237. MR 1202390 Zbl 0813.57013 article

[60] W. D. Neumann and L. V. Thanh: “On irregular links at infinity of algebraic plane curves,” Math. Ann. 295 : 2 (1993), pp. 239–244. MR 1202391 Zbl 0789.14031 ArXiv alg-geom/9202008 article

[61] W. D. Neumann and M. Shapiro: “Automatic structures and boundaries for graphs of groups,” Int. J. Algebra Comput. 4 : 4 (1994), pp. 591–616. MR 1313129 Zbl 0832.20051 ArXiv math/9306205 article

We study the synchronous and asynchronous automatic structures on the fundamental group of a graph of groups in which each edge group is finite. Up to a natural equivalence relation, the set of biautomatic structures on such a graph product bijects to the product of the sets of biautomatic structures on the vertex groups. The set of automatic structures is much richer. Indeed, it is dense in the infinite product of the sets of automatic structures of all conjugates of the vertex groups. We classify these structures by a class of labelled graphs which “mimic” the underlying graph of the graph of groups. Analogous statements hold for asynchronous automatic structures. We also discuss the boundaries of these structures.

[62] W. D. Neumann and M. Shapiro: “Automatic structures, rational growth, and geometrically finite hyperbolic groups,” Invent. Math. 120 : 2 (1995), pp. 259–287. MR 1329042 Zbl 0831.20041 ArXiv math/9401201 article

We show that the set \( \mathbf{S}\mathfrak{U}(G) \) of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group \( G \) is dense in the product of the sets \( \mathbf{S}\mathfrak{U}(P) \) over all maximal parabolic subgroups \( P \). The set \( \mathbf{BS}\mathfrak{U}(G) \) of equivalence classes of biautomatic structures on \( G \) is isomorphic to the product of the sets \( \mathbf{BS}\mathfrak{U}(P) \) over the cusps (conjugacy classes of maximal parabolic subgroups) of \( G \). Each maximal parabolic \( P \) is a virtually abelian group, so \( \mathbf{S}\mathfrak{U}(P) \) and \( \mathbf{BS}\mathfrak{U}(P) \) were computed in [Neumann and Shapiro 1992].

We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics for \( G \) is regular. Moreover, the growth function of \( G \) with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.

[63] W. D. Neumann and J. Yang: “Invariants from triangulations of hyperbolic 3-manifolds,” Electron. Res. Announc. Am. Math. Soc. 1 : 2 (1995), pp. 72–79. MR 1350682 Zbl 0851.57013 article

[64] W. D. Neumann and J. Yang: “Rationality problems for \( K \)-theory and Chern–Simons invariants of hyperbolic 3-manifolds,” Enseign. Math. (2) 41 : 3–4 (1995), pp. 281–296. MR 1365848 Zbl 0861.57022 article

[65] W. D. Neumann: “Kleinian groups generated by rotations,” pp. 251–256 in Groups (Pusan, Korea, 18–25 August 1994). Edited by A. C. Kim and D. L. Johnson. de Gruyter (Berlin), 1995. MR 1476967 Zbl 0872.20045 ArXiv math/9712228 incollection

[66] W. D. Neumann and L. Reeves: “Regular cocycles and biautomatic structures,” Int. J. Algebra Comput. 6 : 3 (1996), pp. 313–324. MR 1404809 Zbl 0928.20028 ArXiv math/9411203 article

[67] W. D. Neumann and M. Shapiro: A short course in geometric group theory, 1996. online notes. Notes for the ANU Workshop January/February 1996. misc

[68] W. D. Neumann: “Commensurability and virtual fibration for graph manifolds,” Topology 36 : 2 (1997), pp. 355–378. MR 1415593 Zbl 0872.57021 article

[69] W. D. Neumann and L. Reeves: “Central extensions of word hyperbolic groups,” Ann. Math. (2) 145 : 1 (January 1997), pp. 183–192. MR 1432040 Zbl 0871.20032 ArXiv math/9507201 article

[70] W. D. Neumann and M. Shapiro: “Regular geodesic normal forms in virtually abelian groups,” Bull. Aust. Math. Soc. 55 : 3 (1997), pp. 517–519. MR 1456281 Zbl 0887.20015 ArXiv math/9702203 article

[71] W. D. Neumann and G. A. Swarup: “Canonical decompositions of 3-manifolds,” Geom. Topol. 1 (1997), pp. 21–40. MR 1469066 Zbl 0886.57009 ArXiv math/9712227 article

[72] W. D. Neumann and J. Yang: Rationality problems for Chern–Simons invariants. Preprint, December 1997. ArXiv math/9712225 techreport

[73] W. D. Neumann and P. Norbury: “Nontrivial rational polynomials in two variables have reducible fibres,” Bull. Aust. Math. Soc. 58 : 3 (1998), pp. 501–503. MR 1662136 Zbl 0946.14038 ArXiv math/9805093 article

[74] W. D. Neumann: “Hilbert’s 3rd problem and invariants of 3-manifolds,” pp. 383–411 in The Epstein birthday schrift. Edited by I. Rivin, C. Rourke, and C. Series. Geometry and Topology Monographs 1. Geometry and Topology Publishers (Coventry, UK), 1998. Dedicated to David Epstein on the occasion of his 60th birthday. MR 1668316 Zbl 0902.57013 ArXiv math/9712226 incollection

This paper is an expansion of my lecture for David Epstein’s birthday, which traced a logical progression from ideas of Euclid on subdividing polygons to some recent research on invariants of hyperbolic 3-manifolds. This “logical progression” makes a good story but distorts history a bit: the ultimate aims of the characters in the story were often far from 3-manifold theory.

We start in section 1 with an exposition of the current state of Hilbert’s 3rd problem on scissors congruence for dimension 3. In section 2 we explain the relevance to 3-manifold theory and use this to motivate the Bloch group via a refined “orientation sensitive” version of scissors congruence. This is not the historical motivation for it, which was to study algebraic \( K \)-theory of \( \mathbb{C} \). Some analogies involved in this “orientation sensitive” scissors congruence are not perfect and motivate a further refinement in Section 4. Section 5 ties together various threads and discusses some questions and conjectures.

David Bernard Alper Epstein	Related
Colin P. Rourke	Related	Volume
Caroline Mary Series	Related

[75] W. D. Neumann and J. Yang: “Bloch invariants of hyperbolic 3-manifolds,” Duke Math. J. 96 : 1 (1999), pp. 29–59. MR 1663915 Zbl 0943.57008 ArXiv math/9712224 article

We define an invariant \( \beta(M) \) of a finite volume hyperbolic 3-manifold \( M \) in the Bloch group \( \mathscr{B}(\mathbb{C}) \) and show it is determined by the simplex parameters of any degree one ideal triangulation of \( M \). We show \( \beta(M) \) lies in a subgroup of \( \mathscr{B}(\mathbb{C}) \) of finite \( \mathbb{Q} \)-rank determined by the invariant trace field of \( M \). Moreover, the Chern–Simons invariant of \( M \) is determined modulo rationals by \( \beta(M) \). This leads to a simplicial formula and rationality results for the Chern–Simons invariant which appear elsewhere.

Generalizations of \( \beta(M) \) are also described, as well as several interesting examples. An appendix describes a scissors congruence interpretation of \( \mathscr{B}(\mathbb{C}) \).

[76] W. D. Neumann: “Conway polynomial of a fibered solvable link,” J. Knot Theor. Ramif. 8 : 4 (1999), pp. 505–509. MR 1697386 Zbl 0941.57017 article

[77] W. D. Neumann: “Irregular links at infinity of complex affine plane curves,” Quart. J. Math. Oxford Ser. (2) 50 : 199 (1999), pp. 301–320. MR 1706321 Zbl 0958.32030 article

[78] Geometric group theory down under (Canberra, 14–19 July 1996). Edited by J. Cossey, C. F. Miller, III, W. D. Neumann, and M. Shapiro. de Gruyter (Berlin), 1999. MR 1714835 Zbl 0910.00040 book

John Cossey	Related
Michael David Shapiro	Related
Charles Frederick Miller, III	Related

[79] C. F. Miller, III, W. D. Neumann, and G. A. Swarup: “Some examples of hyperbolic groups,” pp. 195–202 in Geometric group theory down under (Canberra, 14–19 July 1996). Edited by J. Cossey, C. F. Miller, III, W. D. Neumann, and M. Shapiro. de Gruyter (Berlin), 1999. MR 1714846 Zbl 0955.20027 incollection

Charles Frederick Miller, III	Related
Gadde A. Swarup	Related
Michael David Shapiro	Related
John Cossey	Related

[80] W. D. Neumann and M. Shapiro: “Automatic structures on central extensions,” pp. 261–280 in Geometric group theory down under (Canberra, 14–19 July 1996). Edited by J. Cossey, C. F. Miller, III, W. D. Neumann, and M. Shapiro. de Gruyter (Berlin), 1999. MR 1714849 Zbl 1114.20305 incollection

John Cossey	Related
Michael David Shapiro	Related
Charles Frederick Miller, III	Related

[81] Low dimensional topology (Eger, Hungary, 29 July–2 August 1996 and Budapest, 3–14 August 1998). Edited by K. Böröczky, Jr., W. Neumann, and A. Stipsicz. Bolyai Society Mathematical Studies 8. János Bolyai Mathematical Society (Budapest), 1999. MR 1747267 Zbl 0938.57002 book

András Stipsicz	Related
Károly J. Böröczky	Related

[82] W. D. Neumann: “Notes on geometry and 3-manifolds,” pp. 191–267 in Low dimensional topology (Eger, Hungary, 29 July–2 August 1996 and Budapest, 3–14 August 1998). Edited by K. Böröczky, Jr., W. Neumann, and A. Stipsicz. Bolyai Society Mathematical Studies 8. János Bolyai Mathematical Society (Budapest), 1999. With appendices by Paul Norbury. MR 1747270 Zbl 0944.57012 incollection

Paul Timothy Norbury	Related
András Stipsicz	Related
Károly J. Böröczky	Related

[83] W. D. Neumann and P. Norbury: “Vanishing cycles and monodromy of complex polynomials,” Duke Math. J. 101 : 3 (2000), pp. 487–497. MR 1740685 Zbl 0978.32030 article

In this paper we describe the trivial summand for monodromy around a fibre of a polynomial map \( \mathbb{C}^n\to\mathbb{C} \), generalising and clarifying work of Artal Bartolo, Cassou-Noguès and Dimca [1998], who proved similar results under strong restrictions on the homology of the general fibre and singularities of the other fibres. They also showed a polynomial map \( f:\mathbb{C}^2\to \mathbb{C} \) has trivial global monodromy if and only if it is “rational of simple type” in the terminology of Miyanishi and Sugie. We refine this result and correct the Miyanishi–Sugie classification of such polynomials, pointing out that there are also non-isotrivial examples.

[84] D. Coulson, O. A. Goodman, C. D. Hodgson, and W. D. Neumann: “Computing arithmetic invariants of 3-manifolds,” Experiment. Math. 9 : 1 (2000), pp. 127–152. MR 1758805 Zbl 1002.57044 article

David Eric Coulson	Related
Oliver Andreas Goodman	Related
Craig David Hodgson	Related

[85] W. D. Neumann and P. Norbury: “Unfolding polynomial maps at infinity,” Math. Ann. 318 : 1 (2000), pp. 149–180. MR 1785580 Zbl 1005.32021 ArXiv math/9910054 article

[86] W. D. Neumann and P. G. Wightwick: “Algorithms for polynomials in two variables,” pp. 219–235 in Combinatorial and computational algebra (Hong Kong, 24–29 May 1999). Edited by K. Y. Chan, A. A. Mikhalev, M.-K. Siu, J.-T. Yu, and E. I. Zelmanov. Contemporary Mathematics 264. American Mathematical Society (Providence, RI), 2000. MR 1800698 Zbl 1022.14019 ArXiv math/9911104 incollection

Man Keung Siu	Related
Penelope G. Wightwick	Related

[87] W. D. Neumann: “Immersed and virtually embedded \( \pi_1 \)-injective surfaces in graph manifolds,” Algebr. Geom. Topol. 1 : 1 (2001), pp. 411–426. MR 1852764 Zbl 0979.57007 ArXiv math/9901085 article

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed \( \pi_1 \)-injective surface of negative Euler characteristic. We also determine the class of closed graph manifolds which have no finite cover containing an embedded such surface. This is a larger class. Thus, manifolds \( M^3 \) exist which have immersed \( \pi_1 \)-injective surfaces of negative Euler characteristic, but no such surface is virtually embedded (finitely covered by an embedded surface in some finite cover of \( M^3 \)).

[88] W. D. Neumann and J. Wahl: “Universal Abelian covers of surface singularities,” pp. 181–190 in Trends in singularities. Edited by A. Libgober and M. Tibăr. Trends in Mathematics. Birkhäuser (Basel), 2002. MR 1900786 Zbl 1072.14502 ArXiv math/0110167 incollection

Anatoly S. Libgober	Related
Jonathan Michael Wahl	Related

[89] W. D. Neumann and P. Norbury: “Rational polynomials of simple type,” Pac. J. Math. 204 : 1 (2002), pp. 177–207. MR 1905197 Zbl 1055.14062 ArXiv math/0008080 article

[90] N. Brady, J. P. McCammond, B. Mühlherr, and W. D. Neumann: “Rigidity of Coxeter groups and Artin groups,” pp. 91–109 in Proceedings of the conference on geometric and combinatorial group theory, part 1 (Haifa, Israel, 13–21 June 2000), published as Geom. Dedicata 94 : 1. Issue edited by L. Mosher and M. Sageev. Springer Netherlands (Dordrecht), 2002. MR 1950875 Zbl 1031.20035 incollection

Jonathan Paul McCammond	Related
Bernhard Matthias Mühlherr	Related
Michah El-Yakim Sageev	Related
Noel Patrick Brady	Related

[91] W. D. Neumann and P. Norbury: “The Orevkov invariant of an affine plane curve,” Trans. Am. Math. Soc. 355 : 2 (2003), pp. 519–538. MR 1932711 Zbl 1056.14084 ArXiv math/0110286 article

[92] W. D. Neumann and J. Wahl: “Universal abelian covers of quotient-cusps,” Math. Ann. 326 : 1 (2003), pp. 75–93. MR 1981612 Zbl 1032.14010 ArXiv math/0101251 article

[93] J. H. Manton and W. D. Neumann: “Totally blind channel identification by exploiting guard intervals,” Systems Control Lett. 48 : 2 (2003), pp. 113–119. MR 2011871 Zbl 1134.94303 article

Blind identification techniques estimate the impulse response of a channel by exploiting known finite alphabet or statistical properties of the transmitted symbols. Alternatively, oversampling the output is known to introduce dependencies also exploitable for channel identification. This paper proves the feasibility of estimating the channel by relying instead on the short sequences of zeros, known as guard intervals or zero padding, introduced between transmitted blocks by a number of communication protocols. Since no property of the transmitted information symbols is assumed, the method is called totally blind channel identification. It is proved that totally blind channel identification requires only two received blocks to estimate the channel.

[94] W. D. Neumann: “Topology of hypersurface singularities,” pp. 727–736 in E. Kähler: Mathematische werke [Mathematical works]. Edited by R. Berndt and O. Riemenschneider. Walter de Gruyter (Berlin), 2003. Zbl 1365.14053 ArXiv 1706.04386 incollection

[95] W. D. Neumann: “Extended Bloch group and the Cheeger–Chern–Simons class,” Geom. Topol. 8 : 1 (2004), pp. 413–474. MR 2033484 Zbl 1053.57010 ArXiv math/0307092 article

[96] J. H. Manton, W. D. Neumann, and P. T. Norbury: “On the algebraic identifiability of finite impulse response channels driven by linearly precoded signals,” Systems Control Lett. 54 : 2 (2005), pp. 125–134. MR 2109579 Zbl 1129.93358 article

It is common in wireless communications to perform some form of linear precoding operation on the source signal prior to transmission over a channel. Although the traditional reason for precoding is to improve the performance of the communication system, this paper draws attention to the fact that the receiver can identify the impulse response of the channel without any prior knowledge of the transmitted signal simply by solving a system of polynomial equations. Since different precoders lead to different systems of equations, this paper addresses the fundamental issue of determining which classes of linear precoders lead to a system of equations having a unique solution. In doing so, basic properties of polynomial equations which are useful for studying other identifiability issues commonly encountered in engineering and the applied sciences are presented.

Paul Timothy Norbury	Related
Jonathan H. Manton	Related

[97] W. D. Neumann and J. Wahl: “Complete intersection singularities of splice type as universal abelian covers,” Geom. Topol. 9 : 2 (2005), pp. 699–755. MR 2140991 Zbl 1087.32017 ArXiv math/0407287 article

It has long been known that every quasi-homogeneous normal complex surface singularity with \( \mathbb{Q} \)-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called “splice type singularities,” which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with \( \mathbb{Q} \)-homology sphere links, called “splice-quotient singularities.” According to the Main Theorem, splice-quotients realize a large portion of the possible topologies of singularities with \( \mathbb{Q} \)-homology sphere links. As quotients of complete intersections, they are necessarily \( \mathbb{Q} \)-Gorenstein, and many \( \mathbb{Q} \)-Gorenstein singularities with \( \mathbb{Q} \)-homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with \( \mathbb{Q} \)-homology sphere links are splice-quotients. A recent preprint of T. Okuma presents confirmation of this conjecture.

[98] W. D. Neumann and J. Wahl: “Complex surface singularities with integral homology sphere links,” Geom. Topol. 9 : 2 (2005), pp. 757–811. MR 2140992 Zbl 1087.32018 ArXiv math/0301165 article

While the topological types of normal surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in a previous paper that many of them can be realized as complete intersection singularities of “splice type,” generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. In this paper we prove CIC for a large class of splice type singularities. The CIC suggests (and is motivated by the idea) that the Milnor fiber of a complete intersection singularity with homology sphere link \( \Sigma \) should be a 4-manifold canonically associated to \( \Sigma \). We propose, and verify in a non-trivial case, a stronger conjecture than the CIC for splice type complete intersections: a precise topological description of the Milnor fiber. We also point out recent counterexamples to some overly optimistic earlier conjectures.

[99] W. M. Goldman and W. D. Neumann: “Homological action of the modular group on some cubic moduli spaces,” Math. Res. Lett. 12 : 4 (2005), pp. 575–591. MR 2155233 Zbl 1087.57001 ArXiv math/0402039 article

We describe the action of the automorphism group of the complex cubic \[ x^2 + y^2 + z^2 - xyz - 2 \] on the homology of its fibers. This action includes the action of the mapping class group of a punctured torus on the subvarieties of its \( \mathrm{SL}(2,\mathbb{C}) \) character variety given by fixing the trace of the peripheral element (so-called “relative character varieties”). This mapping class group is isomorphic to \( \mathrm{PGL}(2,\mathbb{Z}) \). We also describe the corresponding mapping class group action for the four-holed sphere and its relative \( \mathrm{SL}(2,\mathbb{C}) \) character varieties, which are fibers of deformations \[ x^2 + y^2 + z^2 - xyz - 2 - Px - Qy - Rz \] of the above cubic. The 2-congruence subgroup \( \mathrm{PGL}(2,\mathbb{Z})_{(2)} \) still acts on these cubics and is the full automorphism group when \( P \), \( Q \), \( R \) are distinct.

[100] W. D. Neumann: “Graph 3-manifolds, splice diagrams, singularities,” pp. 787–817 in Singularity theory (Marseille, France, 24 January–25 February 2005). Edited by D. Chéniot, N. Dutertre, C. Murolo, D. Trotman, and A. Pichon. World Scientific (Hackensack, NJ), 2007. Conference dedicated to Jean-Paul Brasselet on his 60th birthday. MR 2342940 Zbl 1155.32019 incollection

Anne Pichon	Related
Jean-Paul Brasselet	Related

[101] W. D. Neumann and A. Pichon: “Complex analytic realization of links,” pp. 231–238 in Intelligence of low dimensional topology 2006 (Hiroshima, Japan, 22–26 July 2006). Edited by J. S. Carter, S. Kamada, L. H. Kauffman, A. Kawauchi, and T. Kohno. Series on Knots and Everything 40. World Scientific (Hackensack, NJ), 2007. MR 2371730 Zbl 1146.32013 ArXiv math/0610348 incollection

Anne Pichon	Related
Akio Kawauchi	Related
Louis H. Kauffman	Related
Toshitake Kohno	Related

[102] C. J. Leininger, D. B. McReynolds, W. D. Neumann, and A. W. Reid: “Length and eigenvalue equivalence,” Int. Math. Res. Not. 2007 : 24 (2007). Article no. rnm135, 24 pp. MR 2377017 Zbl 1158.53032 ArXiv math/0606343 article

David Ben McReynolds	Related
Christopher J. Leininger	Related
Alan William Reid	Related

[103] W. D. Neumann: A short proof that positive generation implies the Hanna Neumann Conjecture. Preprint, February 2007. ArXiv math/0702395 techreport

[104] J. A. Behrstock and W. D. Neumann: “Quasi-isometric classification of graph manifold groups,” Duke Math. J. 141 : 2 (February 2008), pp. 217–240. MR 2376814 Zbl 1194.20045 ArXiv math/0604042 article

[105] L. Birbrair, A. Fernandes, and W. D. Neumann: “Bi-Lipschitz geometry of weighted homogeneous surface singularities,” Math. Ann. 342 : 1 (2008), pp. 139–144. MR 2415318 Zbl 1153.14003 ArXiv 0704.2041 article

Alexandre César Gurgel Fernandes	Related
Lev B. Birbrair	Related

[106] S. Nelson and W. D. Neumann: “The 2-generalized knot group determines the knot,” Commun. Contemp. Math. 10 : supp01 (2008), pp. 843–847. To the memory of Xiao-Song Lin. MR 2468367 Zbl 1176.57012 ArXiv 0804.0807 article

Victor Samuel Nelson	Related
Xiao-Song Lin	Related

[107] W. D. Neumann and J. Yang: \( \mu \)-constancy does not imply constant bi-Lipschitz type. Preprint, September 2008. ArXiv 0809.0845 techreport

[108] J. A. Behrstock, T. Januszkiewicz, and W. D. Neumann: “Commensurability and QI classification of free products of finitely generated abelian groups,” Proc. Am. Math. Soc. 137 : 3 (2009), pp. 811–813. MR 2457418 Zbl 1183.20025 ArXiv 0712.0569 article

Jason Alan Behrstock	Related
Tadeusz Januszkiewicz	Related

[109] L. Birbrair, A. Fernandes, and W. D. Neumann: “Bi-Lipschitz geometry of complex surface singularities,” Geom. Dedicata 139 (2009), pp. 259–267. MR 2481850 Zbl 1164.32005 ArXiv 0804.0194 article

Alexandre César Gurgel Fernandes	Related
Lev B. Birbrair	Related

[110] W. D. Neumann and J. Wahl: “The end curve theorem for normal complex surface singularities,” J. Eur. Math. Soc. 12 : 2 (2010), pp. 471–503. MR 2608949 Zbl 1204.32019 ArXiv 0804.4644 article

We prove the “End Curve Theorem,” which states that a normal surface singularity \( (X,o) \) with rational homology sphere link \( \Sigma \) is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree.

An “end curve function” is an analytic function \( (X,o)\to (\mathbb{C},0) \) whose zero set intersects \( \Sigma \) in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf.

A “splice quotient singularity” \( (X,o) \) is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in \( \mathbb{C}^t \), where \( t \) is the number of leaves in the resolution graph for \( (X,o) \), together with an explicit description of the covering transformation group.

Among the immediate consequences of the End Curve Theorem are the previously known results: \( (X,o) \) is a splice quotient if it is weighted homogeneous [Neumann 1981], or rational or minimally elliptic [Okuma 2004].

[111] J. A. Behrstock, T. Januszkiewicz, and W. D. Neumann: “Quasi-isometric classification of some high dimensional right-angled Artin groups,” Groups Geom. Dyn. 4 : 4 (2010), pp. 681–692. MR 2727658 Zbl 1226.20033 ArXiv 0906.4519 article

Jason Alan Behrstock	Related
Tadeusz Januszkiewicz	Related

[112] W. D. Neumann: “On Leighton’s graph covering theorem,” Groups Geom. Dyn. 4 : 4 (2010), pp. 863–872. MR 2727669 Zbl 1210.05113 ArXiv 0906.2496 article

[113] L. Birbrair, A. Fernandes, and W. D. Neumann: “Separating sets, metric tangent cone and applications for complex algebraic germs,” Selecta Math. (N.S.) 16 : 3 (2010), pp. 377–391. MR 2734336 Zbl 1200.14010 ArXiv 0905.4312 article

Alexandre César Gurgel Fernandes	Related
Lev B. Birbrair	Related

[114] L. Birbrair, A. Fernandes, and W. D. Neumann: “On normal embedding of complex algebraic surfaces,” pp. 17–22 in Real and complex singularities (São Carlos, Brazil, 27 July–2 August 2008). Edited by M. Manoel, M. C. Romero Fuster, and C. T. C. Wall. London Mathematical Society Lecture Note Series 380. Cambridge University Press, 2010. Dedicated to our friends Maria (Cidinha) Ruas and Terry Gaffney in connection to their 60th birthdays. MR 2759086 Zbl 1215.14057 ArXiv 0901.0030 incollection

Miriam Garcia Manoel	Related
C. T. C. Wall	Related
Maria Aparecida Soares Ruas	Related
Lev B. Birbrair	Related
Alexandre César Gurgel Fernandes	Related
Terence James Gaffney	Related
María del Carmen Romero Fuster	Related

[115] Low-dimensional topology and number theory (Oberwolfach, Germany, 15–21 August 2012), published as Oberwolfach Rep. 7 : 3. Issue edited by P. E. Gunnells, W. D. Neumann, A. S. Sikora, and D. B. Zagier. EMS Press (Berlin), 2010. MR 3156734 Zbl 1209.00056 book

Adam Stanislaw Sikora	Related
Paul Edward Gunnells	Related
Don Bernard Zagier	Related

[116] A. Némethi, W. D. Neumann, and A. Pichon: “Principal analytic link theory in homology sphere links,” pp. 377–387 in Topology of algebraic varieties and singularities: Invited papers of the conference in honor of Anatoly Libgober’s 60th birthday (Jaca, Spain, 22–26 June 2009). Edited by J. I. Cogolludo-Agustín and E. Hironaka. Contemporary Mathematics 538. American Mathematical Society, 2011. MR 2777831 Zbl 1272.32029 ArXiv 0909.1348 incollection

András Némethi	Related
Anatoly S. Libgober	Related
Anne Pichon	Related

[117] W. D. Neumann: “Realizing arithmetic invariants of hyperbolic 3-manifolds,” pp. 233–246 in Interactions between hyperbolic geometry, quantum topology and number theory (New York, 3–13 and 15–19 June 2009). Edited by A. Champanerkar, O. Dasbach, E. Kalfagianni, I. Kofman, W. Neumann, and N. Stoltzfus. Contemporary Mathematics 541. American Mathematical Society (Providence, RI), 2011. MR 2796636 Zbl 1237.57013 ArXiv 1108.0062 incollection

These are notes based on the course of lectures on arithmetic invariants of hyperbolic manifolds given at the workshop associated with the last of three “Volume Conferences,” held at Columbia University, LSU Baton Rouge, and Columbia University respectively in March 2006, May/June 2007, June 2009.

The first part of the lecture series was expository, and since most of the material is readily available elsewhere, we move rapidly over it here (the very first lecture was a rapid introduction to algebraic number theory, here compressed to less than 2 pages, but hopefully sufficient for the topologist who has never had a course in algebraic number theory). Section 2 on arithmetic invariants has some new material, while Section 3 describes a question that Alan Reid and the author first asked about 20 years ago, and describes a very tentative approach. It is here promoted to a conjecture, in part because the author believes he is safe from contradiction in his lifetime.

Ilya Samuil Kofman	Related
Efstratia Kalfagianni	Related
Abhijit Ashok Champanerkar	Related
Oliver T. Dasbach	Related
Neal W. Stoltzfus	Related

[118] Interactions between hyperbolic geometry, quantum topology and number theory (New York, 3–13 and 15–19 June 2009). Edited by A. Champanerkar, O. Dasbach, E. Kalfagianni, I. Kofman, W. Neumann, and N. Stoltzfus. Contemporary Mathematics 541. American Mathematical Society (Providence, RI), 2011. MR 2797089 Zbl 1214.00022 book

Ilya Samuil Kofman	Related
Efstratia Kalfagianni	Related
Abhijit Ashok Champanerkar	Related
Oliver T. Dasbach	Related
Neal W. Stoltzfus	Related

[119] A. Champanerkar, D. Futer, I. Kofman, W. Neumann, and J. S. Purcell: “Volume bounds for generalized twisted torus links,” Math. Res. Lett. 18 : 6 (2011), pp. 1097–1120. MR 2915470 Zbl 1271.57008 ArXiv 1007.2932 article

Ilya Samuil Kofman	Related
Jessica S. Purcell	Related
Abhijit Ashok Champanerkar	Related
David Futer	Related

[120] J. A. Behrstock and W. D. Neumann: “Quasi-isometric classification of non-geometric 3-manifold groups,” J. Reine Angew. Math. 2012 : 669 (2012), pp. 101–120. MR 2980453 Zbl 1252.57001 ArXiv 1001.0212 article

[121] W. D. Neumann and A. Pichon: Lipschitz geometry of complex surfaces: Analytic invariants and equisingularity. Technical report, November 2012. ArXiv 1211.4897 techreport

[122] L. Birbrair, W. D. Neumann, and A. Pichon: “The thick-thin decomposition and the bilipschitz classification of normal surface singularities,” Acta Math. 212 : 2 (2014), pp. 199–256. MR 3207758 Zbl 1303.14016 ArXiv 1105.3327 article

We describe a natural decomposition of a normal complex surface singularity \( (X,0) \) into its “thick” and “thin” parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts.

By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of \( (X,0) \) in terms of its topology and a finite list of numerical bilipschitz invariants.

Anne Pichon	Related
Lev B. Birbrair	Related

[123] L. Birbrair, A. Fernandes, V. Grandjean, and D. O’Shea: “Choking horns in Lipschitz geometry of complex algebraic varieties,” J. Geom. Anal. 24 : 4 (2014), pp. 1971–1981. Appendix by Walter D. Neumann. MR 3261728 Zbl 1307.14003 article

The paper studies the Lipschitz geometry of germs of complex algebraic varieties and introduces the notion of a choking horn. A choking horn is a family of cycles on an algebraic variety with the property that the cycles cannot be boundaries of nearby chains. The presence of choking horns is an obstruction to metric conicalness and the authors use this to prove that some classical isolated hypersurface singularities are not metrically conical. They also show that there exist countably infinitely many singular varieties, which are locally homeomorphic, but not locally subanalytically bi-Lipschitz equivalent with respect to the inner metric. The Appendix by W. Neumann uses separating sets to provide another example of the same phenomenon.

Vincent Grandjean	Related
Donal Bartholomew O'Shea	Related
Lev B. Birbrair	Related
Alexandre César Gurgel Fernandes	Related

[124] W. D. Neumann and A. Pichon: “Lipschitz geometry of complex curves,” J. Singul. 10 (2014), pp. 225–234. MR 3300297 Zbl 1323.14003 ArXiv 1302.1138 article

[125] Low-dimensional topology and number theory (Oberwolfach, Germany, 17–23 August 2014), published as Oberwolfach Rep. 11 : 3. Issue edited by P. E. Gunnells, W. D. Neumann, A. S. Sikora, and D. B. Zagier. EMS Press (Berlin), 2014. MR 3444214 Zbl 1349.00178 book

Adam Stanislaw Sikora	Related
Paul Edward Gunnells	Related
Don Bernard Zagier	Related

[126] W. D. Neumann and A. Tsvietkova: “Intercusp geodesics and the invariant trace field of hyperbolic 3-manifolds,” Proc. Am. Math. Soc. 144 : 2 (2016), pp. 887–896. MR 3430862 Zbl 1360.57020 ArXiv 1402.5582 article

Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these parameters are elements of the invariant trace field of the manifold, providing a connection between the intrinsic geometry of a 3-manifold and its number-theoretic invariants. Further, we explore the question of choosing a minimal collection of arcs and associated parameters to generate the field. We prove that for a tunnel number \( k \) manifold it is enough to choose \( 3k \) specific parameters. For many hyperbolic link complements, this approach allows one to compute the field from a link diagram. We also give examples of infinite families of links where a single parameter can be chosen to generate the field, and the polynomial for it can be constructed from the link diagram as well.

[127] W. D. Neumann and A. Pichon: “Lipschitz geometry does not determine embedded topological type,” pp. 183–195 in Singularities in geometry, topology, foliations and dynamics: A celebration of the 60th birthday of José Seade (Mérida, Mexico, 8–19 December 2014). Edited by J. L. Cisneros-Molina, D. Tráng Lê, M. Oka, and J. Snoussi. Trends in Mathematics. Birkhäuser (Cham, Switzerland), 2017. Dedicated to José Seade for a great occasion. Happy birthday, Pepe. MR 3706220 Zbl 1426.14003 ArXiv 1506.03841 incollection

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in \( (\mathbb{C}^n,0) \). It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type. We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs \( (X_1,0) \) and \( (X_2,0) \) in \( (\mathbb{C}^3,0) \) having the same outer Lipschitz geometry and different embedded topological types. Our pair consist of two superisolated singularities whose tangent cones form an Alexander–Zariski pair having only cusp-singularities. Our result is based on a description of the Lipschitz outer geometry of a superisolated singularity. We also prove that the Lipschitz inner geometry of a superisolated singularity is completely determined by its (non-embedded) topological type, or equivalently by the combinatorial type of its tangent cone.

Anne Pichon	Related
José Antonio Seade Kuri	Related

@incollection {key3706220m,
	AUTHOR = {Neumann, Walter D. and Pichon, Anne},
	TITLE = {Lipschitz geometry does not determine
		 embedded topological type},
	BOOKTITLE = {Singularities in geometry, topology,
		 foliations and dynamics: {A} celebration
		 of the 60th birthday of {J}os\'e {S}eade},
	EDITOR = {Cisneros-Molina, Jos\'e Luis and Tr\'ang
		 L\^e, D\~ung and Oka, Mutsuo and Snoussi,
		 Jawad},
	SERIES = {Trends in Mathematics},
	PUBLISHER = {Birkh\"auser},
	ADDRESS = {Cham, Switzerland},
	YEAR = {2017},
	PAGES = {183--195},
	DOI = {10.1007/978-3-319-39339-1_11},
	NOTE = {(M\'erida, Mexico, 8--19 December 2014).
		 Dedicated to Jos\'e Seade for a great
		 occasion. Happy birthday, Pepe!. ArXiv:1506.03841.
		 MR:3706220. Zbl:1426.14003.},
	ISSN = {2297-0215},
	ISBN = {9783319393384},
}

[128] Low-dimensional topology and number theory (Oberwolfach, Germany, 20–26 August 2017), published as Oberwolfach Rep. 14 : 3. Issue edited by P. E. Gunnells, W. D. Neumann, A. S. Sikora, and D. B. Zagier. EMS Press (Berlin), 2017. MR 3826628 Zbl 1394.00015 book

Adam Stanislaw Sikora	Related
Paul Edward Gunnells	Related
Don Bernard Zagier	Related

[129] W. D. Neumann: Non-isomorphism of categories of algebras. Preprint, August 2018. ArXiv 1808.06242 techreport

[130] A. Fernandes and J. E. Sampaio: “Tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded,” Int. Math. Res. Not. 2019 : 15 (August 2019), pp. 4880–4897. Appendix by Anne Pichon and Walter D. Neumann. MR 3988673 Zbl 1457.32014 ArXiv 1705.00038 article

José Edson Sampaio	Related
Anne Pichon	Related
Alexandre César Gurgel Fernandes	Related

[131] W. D. Neumann, H. M. Pedersen, and A. Pichon: “A characterization of Lipschitz normally embedded surface singularities,” J. Lond. Math. Soc. (2) 101 : 2 (2020), pp. 612–640. MR 4093968 Zbl 1441.14015 ArXiv 1806.11240 article

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We give a necessary and sufficient condition for a normal surface singularity to be Lipschitz normally embedded (LNE), that is, to have bilipschitz equivalent outer and inner metrics. In a partner paper (Neumann, Pedersen and Pichon, J. London Math. Soc. (2020)), we apply it to prove that rational surface singularities are LNE if and only if they are minimal.

Anne Pichon	Related
Helge Møller Pedersen	Related

[132] W. D. Neumann, H. M. Pedersen, and A. Pichon: “Minimal surface singularities are Lipschitz normally embedded,” J. Lond. Math. Soc. (2) 101 : 2 (2020), pp. 641–658. MR 4093969 Zbl 1441.14016 ArXiv 1503.03301 article

Anne Pichon	Related
Helge Møller Pedersen	Related

[133] Introduction to Lipschitz geometry of singularities (Cuernavaca, Mexico, 11–22 June 2018). Edited by W. Neumann and A. Pichon. Lecture Notes in Mathematics 2280. Springer (Cham, Switzerland), 2020. MR 4200092 Zbl 1456.58002 book

[134] W. D. Neumann: “3-manifolds and links of singularities,” pp. 73–86 in Introduction to Lipschitz geometry of singularities (Cuernavaca, Mexico, 11–22 June 2018). Edited by W. Neumann and A. Pichon. Lecture Notes in Mathematics 2280. Springer (Cham, Switzerland), 2020. MR 4200095 Zbl 1457.32071 incollection

[135] W. D. Neumann and J. Wahl: “Orbifold splice quotients and log covers of surface pairs,” J. Singul. 23 (2021), pp. 151–169. MR 4304520 Zbl 07407803 ArXiv 2011.09077 article

A three-dimensional orbifold \( (\Sigma,\gamma_i,n_i) \), where \( \Sigma \) is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair \( (X,C) \), where \( X \) is a normal surface singularity with \( \mathbb{Q}\mathrm{HS} \) link and \( C \) is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, proving the existence of the universal abelian log cover of a pair. A key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with the case where \( C \) is empty and one considers the universal abelian cover, under certain conditions on a resolution graph one can construct pairs and their universal abelian log covers. Such pairs are called orbifold splice quotients.