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Jack S. Calcut

Publications and source records attributed to Jack S. Calcut.

18 recordsLinked to original sources
arXiv2026

Mazur's knot and the Octahedron

Mazur's knot exterior admits a geometric description using a single regular ideal octahedron. The resulting hyperbolic structure is closely related to the Whitehead link exterior through Adams' theorem on thrice-punctured spheres. The same octahedral framework applies to the family of Jester manifolds introduced by Sparks. Using hyperbolic geometry, Thurston's hyperbolic Dehn filling theorem, and Mostow--Prasad rigidity, we prove that Mazur and Jester boundary 3-manifolds are pairwise distinct up to finite ambiguity. Using recent results on systolic geodesics, we remove the remaining finite ambiguity and prove that the boundaries of all Mazur and Jester manifolds are pairwise nonhomeomorphic, regardless of orientation. Consequently, the corresponding compact, contractible $4$-manifolds are pairwise nonhomeomorphic.

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arXiv2026

Structure and classification of torus theta-curves

We study theta-curves embedded in a standard torus in the 3-sphere. We show that each nontrivial torus knot together with an essential arc determines a prime theta-curve, yielding explicit infinite families of prime theta-curves. We compute their constituent knots and identify the structure governing these embeddings, which leads to a complete classification of torus theta-curves up to ambient isotopy and homeomorphism of the 3-sphere. In particular, Kinoshita's theta-curve does not lie on a standard torus.

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arXiv2025

Ends and end cohomology

Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper map. We define reduced end cohomology as the relative end cohomology of a ray-based space. We use those results to prove a version of a theorem of King that computes the reduced end cohomology of an end sum of two manifolds. We include a complete proof of Freudenthal's fundamental theorem on the number of ends of a topological group, and we use our results on dimension-zero end cohomology to prove -- without using transfinite induction -- a theorem of Nöbeling on freeness of certain modules of continuous functions.

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arXiv2023

The end sum of surfaces

End sum is a natural operation for combining two noncompact manifolds and has been used to construct various manifolds with interesting properties. The uniqueness of end sum has been well-studied in dimensions three and higher. We study end sum -- and the more general notion of adding a 1-handle at infinity -- for surfaces and prove uniqueness results. The result of adding a 1-handle at infinity to distinct ends of a surface with compact boundary is uniquely determined by the chosen ends and the orientability of the 1-handle. As a corollary, the end sum of two surfaces with compact boundary is uniquely determined by the chosen ends. Unlike uniqueness results in higher dimensions, which rely on isotopy uniqueness of rays, our results rely fundamentally on a classification of noncompact surfaces.

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arXiv2021

Artin presentations, triangle groups, and 4-manifolds

Gonz{á}lez-Acu{ñ}a showed that Artin presentations characterize closed, orientable $3$-manifold groups. Winkelnkemper later discovered that each Artin presentation determines a smooth, compact, simply-connected $4$-manifold. We utilize triangle groups to find all Artin presentations on two generators that present the trivial group. We then determine all smooth, closed, simply-connected $4$-manifolds with second betti number at most two that appear in Artin presentation theory.

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arXiv2020

Extreme Nonuniqueness of End-Sum

We give explicit examples of pairs of one-ended, open 4-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding the nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theorem is an understanding of this algebra for an end-sum in terms of the algebras of the summands together with ray-fundamental classes determined by the rays used to perform the end-sum. Differing ray-fundamental classes allow us to distinguish the various examples, but only through the subtle theory of infinitely generated abelian groups. An appendix is included which contains the necessary background from that area.

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arXiv2018

On uniqueness of end sums and 1-handles at infinity

For oriented manifolds of dimension at least 4 that are simply connected at infinity, it is known that end summing is a uniquely defined operation. Calcut and Haggerty showed that more complicated fundamental group behavior at infinity can lead to nonuniqueness. The present paper examines how and when uniqueness fails. Examples are given, in the categories TOP, PL and DIFF, of nonuniqueness that cannot be detected in a weaker category (including the homotopy category). In contrast, uniqueness is proved for Mittag-Leffler ends, and generalized to allow slides and cancellation of (possibly infinite) collections of 0- and 1-handles at infinity. Various applications are presented, including an analysis of how the monoid of smooth manifolds homeomorphic to R^4 acts on the smoothings of any noncompact 4-manifold.

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arXiv2016

Double branched covers of theta-curves

We prove a folklore theorem of W. Thurston which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot U is prime if and only if lifting the third arc of the theta-curve to the double branched cover over U produces a prime knot. We apply this result to Kinoshita's theta-curve.

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arXiv2014

Rational Angled Hyperbolic Polygons

We prove that every rational angled hyperbolic triangle has transcendental side lengths and that every rational angled hyperbolic quadrilateral has at least one transcendental side length. Thus, there does not exist a rational angled hyperbolic triangle or quadrilateral with algebraic side lengths. We conjecture that there does not exist a rational angled hyperbolic polygon with algebraic side lengths.

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arXiv2013

Connected sum at infinity and 4-manifolds

We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m at least 3, we construct an infinite family of pairs of m-manifolds on which the connected sum at infinity operation yields distinct manifolds for certain ray choices. We use cohomology algebras at infinity to distinguish these manifolds.

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arXiv2012

Borromean rays and hyperplanes

Three disjoint rays in euclidean 3-space form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.

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arXiv2012

Topological and algebraic pullback functors

We give algebraic equivalents for certain desirable properties of pullback functors on categories of coverings and group sets, namely nullity zero, essential injectivity, and essential surjectivity. Nullity zero turns out to be equivalent to the notion of a contranormal subgroup. We observe a Tannakian-like phenomenon with essential injectivity. Essential surjectivity is intimately related to Zappa-Sz{é}p products. We include several examples, and some open questions.

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arXiv2011

Orbit Spaces of Gradient Vector Fields

We study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit spaces are locally contractible. We also show that the quotient map associated to each such orbit space is a weak homotopy equivalence and has the path lifting property.

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arXiv2011

On fundamental groups of quotient spaces

In classical covering space theory, a covering map induces an injection of fundamental groups. This paper reveals a dual property for certain quotient maps having connected fibers, with applications to orbit spaces of vector fields and leaf spaces in general.

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arXiv2010

Connected sum at infinity and Cantrell-Stallings hyperplane unknotting

We give a general treatment of the somewhat unfamiliar operation on manifolds called Connected Sum at Infinity, or CSI for short. A driving ambition has been to make the geometry behind the well definition and basic properties of CSI as clear and elementary as possible. CSI then yields a very natural and elementary proof of a remarkable theorem of J. C. Cantrell and J. R. Stallings. It asserts unknotting of proper embeddings of euclidean (m-1)-space in euclidean m-space with m not equal to 3, for all three classical manifold categories: topological, piecewise linear, and differentiable. It is one of the few major theorems whose statement and proof can be the same for all three categories. We give it the acronym HLT, which is short for Hyperplane Linearization Theorem. The topological version of the HLT immediately implies B. Mazur's topological Schoenflies theorem. We can thus claim that the Cantrell-Stallings theorem, as we present it, is an enhancement of the topological Schoenflies theorem that has exceptional didactic value. We also prove a classification of multiple codimension 1 hyperplane embeddings in eucldiean m-space for m not equal to 3. Namely, they are classified by countable simplicial trees with one edge for each hyperplane (planar trees for m=2). This result is called the Multiple Hyperplane Linearization Theorem, or MHLT for short. We give an exposition of C. Greathouse's Slab Theorem, and in conclusion some possibly novel proofs of the 2-dimensional MHLT and related results classifying contractible 2-manifolds with boundary.

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arXiv2009

Discreteness and Homogeneity of the Topological Fundamental Group

For a locally path connected topological space, the topological fundamental group is discrete if and only if the space is semilocally simply-connected. While functoriality of the topological fundamental group for arbitrary topological spaces remains an open question, the topological fundamental group is always a homogeneous space.

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