3-manifolds

A **3-manifold** is a topological space that locally resembles Euclidean 3-dimensional space \(\mathbb{R}^3\). More formally, a 3-manifold is a Hausdorff space that is second-countable (any open cover has a countable subcover), and for every point in the manifold, there exists a neighborhood that is homeomorphic to an open subset of \(\mathbb{R}^3\).

The (−2,3,7) pretzel knot is a specific type of pretzel knot, which is a category of knots that can be represented as a sequence of half-twists and crossings. The notation (−2,3,7) specifies the number of crossings and their respective signs in the knot. In this notation: - The "−2" indicates that there are two left-handed (negative) twists.

The Berge knot, also known as the Berge's knot 3_1 or simply the Berge knot, is a specific type of knot in the field of topology and knot theory. It is characterized by its unique structure and properties, which make it an interesting subject of study in mathematics. The Berge knot can be described as a variation of the trefoil knot and is often represented in diagrams with specific crossings.

The term "compression body" can refer to different concepts depending on the context. Here are a few interpretations: 1. **Physics and Mechanics**: In the study of materials and mechanics, a "compression body" may refer to any solid object being subjected to compressive forces. Compressive stress is a force that acts to reduce the volume of the material. When discussing structures or materials, understanding how they behave under compression is important for engineering applications.

Dehn's lemma is a result in geometric topology, specifically in the area of 3-manifolds and the study of surfaces embedded within them. It addresses how certain types of simple homotopies can be related to the topology of surfaces in 3-manifolds.

The Ehrenpreis Conjecture, proposed by is a conjecture in the field of mathematics that relates to the structure of solutions to certain types of partial differential equations (PDEs). Specifically, it addresses solutions of linear PDEs with constant coefficients.

The Ending Lamination Theorem is a significant result in the field of three-dimensional topology, particularly in the study of 3-manifolds and group actions on them. It is primarily associated with the work of Ian Agol and others in the context of geometric topology. In simple terms, the Ending Lamination Theorem provides a way to understand the behavior of hyperbolic 3-manifolds with "infinite area" or those that are "differently closed.

In the context of mathematics, particularly in topology and algebraic geometry, the term "finite type invariant" can refer to certain properties or characteristics associated with topological spaces or algebraic varieties. ### Finite Type Invariant in Algebraic Geometry In algebraic geometry, an invariant of a variety (or a scheme) is said to be of finite type if it can be described in a way that relates to a finite subset of some underlying structure.

Geometric topology is a branch of mathematics that studies the properties of topological spaces and the structures that arise from geometric objects. It primarily focuses on the properties of spaces that are preserved under continuous transformations (homeomorphisms). The field combines ideas from algebraic topology, differential topology, and various geometric considerations. Some key areas of interest in geometric topology include: 1. **3-Manifolds**: A significant portion of geometric topology is devoted to the study of three-dimensional manifolds.

The Hantzsche–Wendt manifold is a specific type of 3-manifold that serves as an example in the study of topology and geometry. It can be characterized as a compact, orientable, triangulated manifold with non-trivial fundamental group. One main feature of the Hantzsche–Wendt manifold is that it can be constructed from 3-dimensional Euclidean space and is related to the theory of solvable Lie groups.

A horosphere is a geometric concept commonly encountered in differential geometry and hyperbolic geometry. It can be thought of as a generalization of the notion of a sphere in hyperbolic space. More formally: 1. **Definition**: In hyperbolic space, a horosphere is defined as the set of points that are at a constant hyperbolic distance from a given point on the boundary at infinity of hyperbolic space.

A hyperbolic 3-manifold is a type of three-dimensional manifold that possesses a geometry modeled on hyperbolic space. Specifically, a hyperbolic 3-manifold is characterized by having a constant negative curvature, which means that its geometric properties are governed by hyperbolic geometry, rather than Euclidean or spherical geometries.

Hyperbolic Dehn surgery is a technique in the study of 3-manifolds, primarily in the field of low-dimensional topology. It involves a process of modifying a given three-dimensional manifold by removing a solid torus and gluing it back in a different way, thus altering the topology of the manifold.

A hyperbolic link in mathematics, particularly in the study of topology and knot theory, refers to a certain type of link (a collection of knots that may be intertwined) that has a hyperbolic structure. This means that the complement of the link in three-dimensional space can be equipped with a Riemannian metric of constant negative curvature.

An **incompressible surface** is a concept from the field of topology, specifically in the study of 3-manifolds. It refers to a two-dimensional surface that cannot be compressed into a simpler form without cutting it. This property is significant in both mathematical theory and applications, such as in knot theory and the study of 3-manifolds.

JSJ decomposition, named after mathematicians William Jaco, Henry Shalen, and William Meier, is a technique used in the field of three-manifold topology. It provides a way to decompose a compact, oriented, irreducible 3-manifold into simpler pieces.

A **Kleinian group** is a type of discontinuous group of isometries of hyperbolic 3-space (denoted as \(\mathbb{H}^3\)).

A lens space is a specific type of three-dimensional manifold that can be thought of as a generalization of the notion of a solid torus. More formally, lens spaces are a class of manifolds that can be defined using the quotient of the 3-sphere \( S^3 \) by a specific action of the group \( \mathbb{Z}/p\mathbb{Z} \), where \( p \) is a positive integer.

The Meyerhoff manifold is a specific type of 3-dimensional manifold that is associated with hyperbolic geometry. It is notable for being an example of a hyperbolic 3-manifold that is particularly well-studied in the field of topology and geometric group theory. The Meyerhoff manifold can be constructed as a quotient of hyperbolic 3-space by a group of isometries.

In the context of mathematics, particularly in topology and differential geometry, a **normal surface** typically refers to a type of surface that is embedded in a three-dimensional space and satisfies certain conditions regarding its curvature and other geometric properties. However, the term "normal surface" may also have specific meanings in different subfields of mathematics, such as in the study of 3-manifolds or algebraic geometry.

A \( P^2 \)-irreducible manifold is a concept from differential topology and algebraic topology, often discussed in the context of 4-manifolds. To understand the term, we first need to break down some components. 1. **4-manifold**: A 4-manifold is a topological space that locally resembles \(\mathbb{R}^4\).

The Picard horn, also known as a Picard trumpet or Picard cone, is a type of mathematical object that arises in the study of topology and algebraic geometry. More specifically, it is a geometric structure that can be formed as a cone over a certain topological space, often related to the concept of a 'horn' in three-dimensional space.

A pleated surface, in the context of geometry and materials science, generally refers to a surface that has been designed with folds or pleats, resembling the folds of fabric in clothing. These surfaces exhibit a series of parallel ridges or valleys that create an aesthetically appealing texture and can serve both functional and decorative purposes. Pleated surfaces can be found in various applications, including: 1. **Fashion Design**: In clothing, pleating is a technique used to create texture and volume.

"Pretzel link" may refer to a few different concepts depending on the context. Here are a couple of possibilities: 1. **Pretzel (Snack)**: In the most common context, a pretzel is a baked bread product, usually shaped into a knot or loop, and often sprinkled with coarse salt. A "link" in this context might refer to a recipe link or a product link associated with pretzels.

The prime decomposition of 3-manifolds is a fundamental concept in the field of 3-manifold topology. It states that any compact connected 3-manifold can be uniquely decomposed into a connected sum of prime 3-manifolds, with the understanding that the connected summands are considered up to homeomorphism. ### Key Concepts: 1. **3-Manifold**: A 3-manifold is a space that locally looks like Euclidean 3-dimensional space.

The Property P conjecture is a concept in the field of mathematical logic and model theory, particularly related to the study of structures and their properties. It specifically deals with structures that are represented by certain kinds of mathematical objects, such as groups, ordered sets, fields, etc. While there are many different contexts in which the term "Property P" could arise, it is often associated with the idea of a certain property, "P", that might be preserved or exhibited under certain operations or transformations.

Ricci flow is a process in differential geometry introduced by mathematician Richard S. Hamilton in 1982. It is a mechanism for deforming the metric of a Riemannian manifold in order to simplify its geometric structure. The primary goal of Ricci flow is to gradually "smooth out" irregularities in the manifold's shape over time.

The Scott core theorem is a result in the field of theoretical computer science, specifically in the areas of domain theory and denotational semantics. It is named after Dana Scott, who made significant contributions to the understanding of computation and programming languages through the development of domain theory. In essence, the Scott core theorem characterizes the way that certain kinds of mathematical structures can be represented and manipulated in a way that is useful for reasoning about computation.

The Seifert-Weber space is a specific type of 3-manifold that can be constructed as a nontrivial example of a Seifert fibered space. It is particularly known for its interesting topological properties. In simpler terms, a Seifert fibered space is a 3-manifold that can be decomposed into a collection of circles (fibers) such that around each fiber, there is a well-defined surface that varies continuously.

The Smith conjecture is a statement in the field of geometric topology and, more specifically, it relates to the structure of 3-manifolds. Proposed by the mathematician Peter B.

A solid Klein bottle is a three-dimensional object that is a higher-dimensional analog of the Klein bottle, which is a non-orientable surface. ### Klein Bottle: The classic Klein bottle can be visualized as a surface that loops back onto itself without any boundaries.

A solid torus is a three-dimensional geometric shape that resembles a doughnut or ring. It is defined as the three-dimensional region that is obtained by taking a two-dimensional disk and revolving it around an axis that is coplanar with the disk but does not intersect it.

A surface bundle over the circle is a type of fiber bundle where the fibers are surfaces and the base space is the circle \( S^1 \).

The Surface Subgroup Conjecture is a conjecture in the field of geometric topology and group theory, particularly related to the study of fundamental groups of 3-manifolds. It states that every finitely generated, word hyperbolic group contains a subgroup that is isomorphic to the fundamental group of a closed surface of genus at least 2.

The geometry and topology of three-manifolds is a rich and complex area of mathematics that deals with understanding the properties and structures of three-dimensional spaces (or manifolds). Here are the key concepts and themes involved: ### Manifolds A **manifold** is a topological space that locally resembles Euclidean space. An **n-manifold** is a space that is locally similar to \( \mathbb{R}^n \).

The Thurston Elliptization Conjecture is a significant statement in the field of topology, particularly concerning 3-manifolds.

The Virtually Haken Conjecture is a conjecture in the field of geometric topology, specifically related to 3-manifolds. It posits that every closed, irreducible 3-manifold that has a fundamental group that is a free product of finitely many non-trivial groups is "virtually Haken." To unpack this, a few definitions are necessary: 1. **Closed 3-manifold**: A 3-manifold that is compact and without boundary.

The Virtually Fibered Conjecture is a conjecture in the field of geometric topology, particularly concerning 3-manifolds. It posits that every aspherical closed irreducible 3-manifold that is not a torus or a connected sum of tori is "virtually fibered." To explain further: - A **3-manifold** is a three-dimensional topological space that locally looks like Euclidean 3-dimensional space.

The Weeks manifold is a specific example of a closed 3-manifold that is often studied in the field of topology and geometric topology. It is particularly noted for its properties in relation to hyperbolic geometry. ### Key Features of the Weeks Manifold: 1. **Closed 3-Manifold**: The Weeks manifold is compact, has no boundary, and can be considered a type of three-dimensional shape.