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Shijie Gu

Publications and source records attributed to Shijie Gu.

15 recordsLinked to original sources
arXiv2026

Kwasik--Schultz manifolds are $\mathcal{Z}$-compactifiable

Kwasik and Schultz constructed two-ended open $4$-manifolds which satisfy the usual finiteness and stability conditions at infinity but do not admit arbitrarily small $1$-neighborhoods. In particular, neither end is collarable, so the manifolds are not completable. We show that the open manifolds associated to their non-desuspendable $C_2$-actions nevertheless admit finite-dimensional compact ANR $\mathcal{Z}$-compactifications. Consequently, there exists a $\mathcal{Z}$-compactifiable open $4$-manifold which is not pseudo-collarable. This answers a question of Guilbault and Tinsley.

math.GT↗
arXiv2026

Busemann G-spaces with convex balls

We prove that any Busemann G-space such that every sufficiently small metric ball is convex is a topological manifold. The key ingredient in the proof is Ivanov's Helly theorem. The appendix contains a counterexample to a question of Berestovskii--Halverson--Repov\v{s}.

math.MG↗
arXiv2026

Finsler structure of Busemann G-spaces

We provide two sufficient conditions for a Busemann G-space to admit a differentiable DC atlas with a continuous Finsler metric, from the viewpoint of comparison geometry. These results generalize previous work on G-spaces with Riemannian curvature bounds, namely the Alexandrov and CAT conditions, to the Finsler setting.

math.DG↗
arXiv2025

Nonembeddable contractible open manifolds arising from Whitehead doubling

We study a family of genus-one contractible open manifolds constructed by iterated Whitehead doubling from a nontrivial knot $K$ and an even half-twist $m$. For each such pair, we prove that the resulting contractible open $3$-manifold $W(K,m)$ does not embed as an open subset of any compact, locally connected and locally $1$-connected metric $3$-space. We also classify this family: $W(K,m)$ is homeomorphic to $W(K',m')$ if and only if $m=m'$ and $K$ is isotopic to $K'$. Thus the knot type and the half-twist parameter form a complete invariant for these nonembeddable contractible open manifolds. The proof combines the topology of ends with JSJ decompositions, hyperbolic pieces, and a rank estimate for iterated Whitehead doubled knot groups. The same method gives infinitely many pairwise non-homeomorphic higher-dimensional examples which embed in no compact, locally connected and locally $1$-connected metric space of the same dimension.

math.GT↗
arXiv2025

Topological regularity of Busemann spaces of nonpositive curvature

We extend the topological results of Lytchak-Nagano and Lytchak-Nagano-Stadler for CAT(0) spaces to the setting of Busemann spaces of nonpositive curvature, i.e., BNPC spaces. We give a characterization of locally BNPC topological manifolds in terms of their links and show that the singular set of a locally BNPC homology manifold is discrete. We also prove that any (globally) BNPC topological 4-manifold is homeomorphic to Euclidean space. Applications include a topological stability theorem for locally BNPC G-spaces. Our arguments also apply to spaces admitting convex geodesic bicombings.

math.DG↗
arXiv2023

On Z-compactifiability of manifolds

In 1976, Chapman and Siebenmann established necessary and sufficient conditions for $\Z$-compactifying Hilbert cube manifolds. Although the corresponding conditions are known to be necessary for a manifold $M^n$ to admit a $\Z$-compactification, it remains open whether they are sufficient. Guilbault and the author proved that they are sufficient for $M^n\times I$, when $n\geq5$. We further explore this question by giving additional hypotheses under which the interval factor can be removed. A retraction defined near the central added set is sufficient; a product-compatible splitting also gives a collar; and an upper-semicontinuous cell-like decomposition gives a splitting-free criterion. We also answer affirmatively a question of Guilbault--Tinsley by showing that: for every $n\ge6$ there is a connected one-ended open PL $n$-manifold which is $\Z$-compactifiable but not pseudo-collarable. Finally, we discuss the additional control needed for applications to universal covers of closed aspherical manifolds.

math.GT↗
arXiv2020

Contractible open manifolds which embed in no compact, locally connected and locally 1-connected metric space

This paper pays a visit to a famous contractible open 3-manifold $W^3$ proposed by R. H. Bing in 1950's. By the finiteness theorem \cite{Hak68}, Haken proved that $W^3$ can embed in no compact 3-manifold. However, until now, the question about whether $W^3$ can embed in a more general compact space such as a compact, locally connected and locally 1-connected metric 3-space was not known. Using the techniques developed in Sternfeld's 1977 PhD thesis \cite{Ste77}, we answer the above question in negative. Furthermore, it is shown that $W^3$ can be utilized to produce counterexamples for every contractible open $n$-manifold ($n\geq 4$) embeds in a compact, locally connected and locally 1-connected metric $n$-space.

math.GT↗
arXiv2019

Characterization of pseudo-collarable manifolds with boundary

In this paper we obtain a complete characterization of pseudo-collarable $n$-manifolds for $n\geq 6$. This extends earlier work by Guilbault and Tinsley to allow for manifolds with noncompact boundary. In the same way that their work can be viewed as an extension of Siebenmann's dissertation that can be applied to manifolds with non-stable fundamental group at infinity, our main theorem can also be viewed as an extension of the recent Gu-Guilbault characterization of completable $n$-manifolds in a manner that is applicable to manifolds whose fundamental group at infinity is not peripherally stable.

math.GT↗
arXiv2018

Compactifications of manifolds with boundary

This paper is concerned with "nice" compactifications of manifolds. Siebenmann's iconic dissertation characterized open manifolds M^m (m>5) compactifiable by addition of a manifold boundary. His theorem extends easily to cases where M^m is noncompact with compact boundary; however, when Bd(M^m) is noncompact, the situation is more complicated. The goal becomes a "completion" of M^m, ie, a compact manifold C^m and a compact subset A such that C^m\A = M^m. Siebenmann did some initial work on this topic, and O'Brien extended that work to an important special case. But, until now, a complete characterization had yet to emerge. We provide such a characterization. Our second main theorem involves Z-compactifications. An open question asks whether a well-known set of conditions laid out by Chapman and Siebenmann guarantee Z-compactifiability for a manifold M^m. We cannot answer that question, but we do show that those conditions are satisfied if and only if M x [0,1] is Z-compactifiable. A key ingredient is the above Manifold Completion Theorem---an application that partly explains our current interest in that topic, and also illustrates the utility of the conditions found in that theorem.

math.GT↗
arXiv2016

A hierarchy for closed n-cell-complements

Let $C$ and $D$ be a pair of crumpled $n$-cubes and $h$ a homeomorphism of $\text{Bd }C$ to $\text{Bd }D$ for which there exists a map $f_h: C\to D$ such that $f_h|\text{Bd }C =h$ and $f_{h}^{-1}(\text{Bd }D)=\text{Bd }C$. In our view the presence of such a triple $(C,D,h)$ suggests that $C$ is "at least as wild as" $D$. The collection $\mathscr{W}_n$ of all such triples is the subject of this paper. If $(C,D,h)\in \mathscr{W}_n$ but there is no homeomorphism such that $D$ is at least as wild as $C$, we say $C$ is "strictly wilder than" $D$. The latter concept imposes a partial order on the collection of crumpled $n$-cubes. Here we study features of these wildness comparisons, and we present certain attributes of crumpled cubes that are preserved by the maps arising when $(C,D,h) \in \mathscr{W}_n$. The effort can be viewed as an initial way of classifying the wildness of crumpled cubes.

math.GT↗
arXiv2014

On the Shrinkable U.S.C. Decomposition Spaces of Spheres

Let $G$ be a u.s.c decomposition of $S^n$, $H_G$ denote the set of nondegenerate elements and $π$ be the projection of $S^n$ onto $S^n/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with ($n-1$)-sphere frontiers which miss $π(H_G)$, and such frontiers satisfies the Mismatch Property. Then this paper shows that this condition implies $S^n/G$ is homeomorphic to $S^n$ ($n\geq 4$). This answers a weakened form of a conjecture asked by Daverman [3, p. 61]. In the case $n=3$, the strong form of the conjecture has an affirmative answer from Woodruff [12].

math.GT↗
arXiv2013

Shrinkability of Decomposition of $S^n$ Having Arbitrarily Small Neighborhoods with ($n-1$)-Sphere Frontiers

Let $G$ be a usc decomposition of $S^n$, $H_G$ denote the set of nondegenerate elements and $π$ be the natural projection of $S^n$ onto $S^n/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with ($n-1$)-sphere frontiers or boundaries which miss $π(H_G)$. If all the arcs are tame in the particular area on the boundary of an $n$-cell $C$ in $S^n$, then this paper shows that this condition implies $S^n/G$ is homeomorphic to $S^n$ ($n\geq 4$). This answers a weak form of a conjecture asked by Daverman [2, p. 61]. In the case of $n=3$, the strong form of the conjecture has an affirmative answer from Woodruff [11].

math.GT↗