Narrow Results

We correct the stability bound in our classification of closed, spin${}^{+}$, topological four-manifolds with fundamental group $\pi$ of cohomological dimension $\le 3$ (up to $s$-cobordism), after stabilization by connected sum with copies of $S^2\times S^2$. If $\pi$ is a right-angled Artin group whose defining graphs have no four-cliques, then the new stability bound is $r\ge max(b_3(\pi ),6)$. The other results of the paper are not affected.

Background: Radial head fractures with comminution and displacement present challenges in achieving optimal treatment outcomes in the long term. This study aims to evaluate the long-term effects of radial head excision (RHE) in patients with Mason type 3 fractures.

Methods: We conducted a retrospective study of patients with a Mason type 3 radial head fracture who underwent primary RHE between January 2010 and January 2020. The primary outcome was the Mayo elbow performance score (MEPS). Additionally, joint stability and arthritis, and the carrying angle were recorded for each patient.

Results: In total, 61 patients (21; 34% female) with a follow-up range of 3–13 years were examined. The mean (SD) total MEPS was 91.8 (9.2). The results were excellent for 46, good for 12 and fair for 3 patients. A model of total MEPS adjusted for sex, age and follow-up time showed a significant effect of patient age on treatment success (p < 0.001). Thirteen patients (21%) showed elbow instability. Six individuals had increased valgus laxity. The mean carrying angle was 19° (range: 11°–27°) on the injured side and 9° (4°–15°) on the uninjured side, t(120) = 12.608, p < 0.001. Overall, 37 patients had degenerative changes in the operative elbow.

Conclusions: Benefits of RHE persist for a long time with predominantly excellent elbow function and minimal complications. An increase in the carrying angle, joint instability and degenerative changes are to be expected. Patient age at the time of the surgery can affect treatment outcomes.

Level of Evidence: Level IV (Therapeutic)

Let N be a closed connected smooth four-manifold with H₁(N; ℤ) = 0. Our main result is the following classification of the set E⁷(N) of smooth embeddings N → ℝ⁷ up to smooth isotopy. Haefliger proved that E⁷(S⁴) together with the connected sum operation is a group isomorphic to ℤ₁₂. This group acts on E⁷(N) by an embedded connected sum. Boéchat and Haefliger constructed an invariant ℵ: E⁷(N) → H₂(N;ℤ) which is injective on the orbit space of this action; they also described im(ℵ). We determine the orbits of the action: for u ∈ im(ℵ) the number of elements in ℵ^-1(u) is GCD (u/2, 12) if u is divisible by 2, or is GCD(u, 3) if u is not divisible by 2. The proof is based on Kreck's modified formulation of surgery.

We investigate the behavior of solutions of the normalized Ricci flow under surgeries of four-manifolds along circles by using Seiberg–Witten invariants. As a by-product, we prove that any pair (α, β) of integers satisfying α + β ≡ 0 (mod 2) can be realized as the Euler characteristic χ and signature τ of infinitely many closed smooth 4-manifolds with negative Perelman's invariants and on which there is no nonsingular solution to the normalized Ricci flows for any initial metric. In particular, this includes the existence theorem of non-Einstein 4-manifolds due to Sambusetti [An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann.311 (1998) 533–547] as a special case.

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddingsN → S^mfinite? In case when the manifold N is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold N is a disjoint union of spheres the answer was given by D. Crowley, S. Ferry and the author in 2011. We consider the next natural case when N is a product of two spheres. In the following theorem, FCS(i, j) ⊂ ℤ² is a specific set depending only on the parity of i and j which is defined in the paper.

Theorem.Assume thatm > 2p + q + 2andm < p + 3q/2 + 2. Then the set of isotopy classes ofC¹-smooth embeddingsS^p × S^q → S^mis infinite if and only if eitherq + 1orp + q + 1is divisible by 4, or there exists a point(x, y)in the setFCS(m - p - q, m - q)such that(m - p - q - 2)x + (m - q - 2)y = m - 3.

Our approach is based on a group structure on the set of embeddings and a new exact sequence, which in some sense reduces the classification of embeddings S^p × S^q → S^m to the classification of embeddings S^p+q ⊔ S^q → S^m and D^p × S^q → S^m. The latter classification problems are reduced to homotopy ones, which are solved rationally.

We investigate multiframings of a closed oriented 3-manifold. We show that multiframings give a geometric realization of the tensor product of the homotopy set of framings and $\mathbb{Q}$. We prove that the Hirzebruch defect defines a bijection from the homotopy set of multiframings to $\mathbb{Q}$ for any connected closed oriented 3-manifold, and we prove that any multiframing defined near the boundary of a compact oriented 3-manifold extends to the bounded manifold.

To establish the proper analgesic method by electroacupuncture (EA) for bovine surgery, the analgesic effect of dorsal and lumbar acupoints, in addition to the combination with dorsal and lumbar acupoints, were investigated in the present study. Four Korean native cattle (two males and two females) and 24 Holstein-Friesian cattle (all females) were used. The experimental animals were divided into four groups according to used acupoints: dorsal acupoint group (Tian Ping [GV-20] and Bai Hui [GV-5]: 7 heads), lumbar acupoint group (Yap Pang 1 [BL-21], Yao Pang 2 [BL-23], Yao Pang 3 [BL-24] and Yao Pang 4 [BL-25]; 5 heads), dorsal-lumbar acupoint group (Yao Pang 1 [BL-21], Yao Pang 2 [BL-23], Yao Pang 3 [BL-24] and Bai Hui [GV-5]; 8 heads) and control group (non-acupoints, the last intercostals space and the femoral area; 3 heads). The acupoints were stimulated with currents of 2–6 V (30 Hz) in dorsal acupoint group, 0.5–2.0 V (30 Hz) in lumbar acupoint group and 0.3–2.5 V (30 Hz) in dorsal-lumbar acupoint group. Recumbency time was 10 seconds to 1 minute (except one case) and induction time of analgesia was approximately 1 to 6 minutes in dorsal acupoint group. Analgesic effect was systemic, including the extremities in dorsal acupoint group. During the EA, the consciousness was evident and blepharo-reaction was still present under EA in dorsal acupoint group. During the surgery, grades of analgesic effect were 6 excellent (6/7, 87.5%) and 1 good (1/7, 14.3%). In addition, induction time for analgesia was about 10 minutes in both lumbar and dorsal-lumbar acupoint groups. Analgesic areas were found in abdominal areas from the last intercostal spaces to the femoral areas, except lower abdomen in lumbar and lumbar-dorsal acupoint groups. The consciousness was evident and standing position was maintained during EA stimulation in contrast to that of dorsal excellent (1/5, 20.0%), 3 good (3/5, 60.0%) and 1 poor (1/5, 20.0%) in the lumbar acupoint group. Additionally, grades of analgesic effect were 4 excellent (4/8, 50.0%), 3 good (3/8, 37.5%) and 1 poor (1/8, 12.5%). On the other hand, pain was present and analgesia was not accomplished under EA stimulation in control group. In conclusion, analgesia by EA was effective with decreasing order of dorsal acupoint > dorsal-lumbar acupoint > lumbar acupoint among groups. It was considered that dorsal acupoint group might be useful for operation with recumbent position, and lumbar and dorsal-lumbar acupoint groups might be proper for operation with standing position.

It is well-known that every 3-manifold M³ may be represented by a framed link (L,c), which indicates the Dehn-surgery from to M³ = M³(L,c); moreover, M³ is the boundary of a PL 4-manifold M⁴ = M⁴(L, c), which is obtained from by adding 2-handles along the framed link (L, c).

In this paper we study the relationships between the above representations and the representation theory of general PL-manifolds by edge-coloured graphs: in particular, we describe how to construct a 5-coloured graph representing M⁴=M⁴(L,c), directly from a planar diagram of (L,c). As a consequence, relations between the combinatorial properties of the link L and both the Heegaard genus of M³=M³(L,c) and the regular genus of M⁴=M⁴(L,c) are obtained.

A classical theorem of R. H. Bing states that a closed connected 3-manifold M is homeomorphic to the 3-sphere if and only if every knot in M is contained in a 3-ball. We give a simple proof of this characterization based on the surgery presentation of 3-manifolds.

The notion of 2-framed three-manifolds is defined. The category of 2-framed cobordisms is described, and used to define a 2-framed three-dimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2-framed three-dimensional TQFT. These data and relations are expressed in the language of surgery.

We give two proofs that the 3-torus is not weakly d-congruent to #³S¹ × S², if d > 2. We study how cohomology ring structure relates to weak congruence. We give an example of three 3-manifolds which are weakly 5-congruent but are not 5-congruent.

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on the same link could give the same three-manifold M.

Let D be a link diagram with n crossings, s_A and s_B be its extreme states and |s_AD| (respectively, |s_BD|) be the number of simple closed curves that appear when smoothing D according to s_A (respectively, s_B). We give a general formula for the sum |s_AD| + |s_BD| for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |s_AD| + |s_BD| = n + 2 - 2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu [On Kauffman brackets, J. Knot Theory Ramifications6(1) (1997) 125–148.]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al. [Almost alternating links, Topology Appl.46(2) (1992) 151–165.], is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram.

In this paper, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules equipped with their Blanchfield forms ϕ, and the modules such that there is a unique isomorphism class of , and we prove that for the other modules , there are infinitely many such classes. We realize all these by explicit knots in ℚ-spheres.

We exhibit a finite set of local moves that connect any two surgery presentations of the same 3-manifold via framed links in S³. The moves are handle-slides and blow-downs/ups of a particular simple kind.

We prove that for particular infinite families of $L$-spaces, arising as branched double covers, the $d$-invariants defined by Ozsváth and Szabó assume arbitrarily large positive and negative values. As a consequence, we generalize a result by Greene and Watson by proving, for every odd number $\mathrm{\Delta }\ge 5$, the existence of infinitely many non-quasi-alternating homologically thin knots with determinant ${\mathrm{\Delta }}^2$, and a result by Hoffman and Walsh concerning the existence of hyperbolic weight $1$ manifolds, that are not surgery on a knot in $S^3$.

Let $X$ and $Y$ be the complementary regions of a closed hypersurface $M$ in $S^4=X{\cup }_{M}Y$. We use the Massey product structure in $H^{\ast }(M;\mathbb{Z})$ to limit the possibilities for $\chi (X)$ and $\chi (Y)$. We show also that if ${\pi }_1(X)\ne 1$ then it may be modified by a 2-knot satellite construction, while if $\chi (X)\le 1$ and ${\pi }_1(X)$ is abelian then ${\beta }_1(M)\le 4$ or ${\beta }_1(M)=6$. Finally we use TOP surgery to propose a characterization of the simplest embeddings of $F\times S^1$.

If $X$ is an orientable, strongly minimal $P{D}_4$-complex and ${\pi }_1(X)$ has one end, then it has no nontrivial locally finite normal subgroup. Hence, if $\pi$ is a 2-knot group, then (a) if $\pi$ is virtually solvable, then either $\pi$ has two ends or $\pi \cong \mathrm{\Phi }$, with presentation $〈a,t|ta=a^{2}t〉$, or $\pi$ is torsion-free and polycyclic of Hirsch length 4 (b) either $\pi$ has two ends, or $\pi$ has one end and the center $\zeta \pi$ is torsion-free, or $\pi$ has infinitely many ends and $\zeta \pi$ is finite, and (c) the Hirsch–Plotkin radical $\sqrt{\pi }$ is nilpotent.

We consider embeddings of 3-manifolds in $S^4$ such that each of the two complementary regions has an abelian fundamental group. In particular, we show that an homology handle $M$ has such an embedding if and only if ${\pi }_1{(M)}^{\prime }$ is perfect, and that the embedding is then essentially unique.

We consider embeddings of 3-manifolds $M$ in $S^4$ such that the two complementary regions $X$ and $Y$ each have nilpotent fundamental group. If $\beta ={\beta }_1(M)$ is odd then these groups are abelian and $\beta \le 3$. In general ${\pi }_1(X)$ and ${\pi }_1(Y)$ have 3-generator presentations, and $\beta \le 6$. We give two examples illustrating our results.