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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 \(\mathbb{R}^3\).

A **4-manifold** is a type of mathematical object studied in the field of topology and differential geometry. In general, an **n-manifold** is a space that locally resembles Euclidean space of dimension \( n \). This means that around every point in a 4-manifold, there exists a neighborhood that is homeomorphic (structurally similar) to an open subset of \( \mathbb{R}^4 \).

In the context of differential geometry and topology, "maps of manifolds" typically refers to smooth or continuous functions that associate points from one manifold to another. Manifolds themselves are mathematical structures that generalize the concept of curves and surfaces to higher dimensions. They can be thought of as "locally Euclidean" spaces, meaning that around any point in a manifold, one can find a neighborhood that looks like Euclidean space.

The term "2-sided" can refer to various concepts depending on the context. Here are a few interpretations: 1. **Physical Objects:** In a physical sense, something that is 2-sided has two distinct sides. This could refer to paper, signs, or any flat object that has a front and a back. 2. **Negotiation:** In the context of negotiation or discussions, a 2-sided approach implies that both parties have the opportunity to express their views, concerns, or proposals.

A 3-manifold is a topological space that locally resembles Euclidean 3-dimensional space. More formally, a space \( M \) is called a 3-manifold if every point in \( M \) has a neighborhood that is homeomorphic (topologically equivalent) to an open subset of \( \mathbb{R}^3 \).

A 4-manifold is a topological space that locally resembles \(\mathbb{R}^4\) (four-dimensional Euclidean space) and is a type of manifold—a fundamental concept in topology and differential geometry. Formally, a 4-manifold \(M\) is a space that is Hausdorff, second countable, and locally homeomorphic to \(\mathbb{R}^4\).

In mathematics, specifically in the field of topology, a **5-manifold** is a topological space that is locally similar to Euclidean space of dimension 5. This means that around every point in the manifold, there exists a neighborhood that is homeomorphic (topologically equivalent) to an open set in \(\mathbb{R}^5\).

Alexander's trick is a technique used in topology, specifically in the study of continuous functions and compactness. It is primarily associated with the construction of continuous maps and the extension of functions. The trick is named after the mathematician James W. Alexander II and is often employed in scenarios where one needs to extend continuous functions from a subspace to a larger space.

The Alexander horned sphere is a classic example in topology, specifically in the study of knot theory and manifold theory. It is constructed by taking a sphere and creating a complex embedding that demonstrates non-standard behavior in three-dimensional space. The construction of the Alexander horned sphere involves a series of increasingly complicated iterations that result in a space that is homeomorphic to the standard 2-sphere but is not nicely embedded in three-dimensional Euclidean space.

The Annulus theorem is a concept in mathematics, particularly in complex analysis and number theory. While the term "Annulus theorem" could refer to different results depending on the context, one notable application relates to properties of holomorphic functions defined on annular regions in the complex plane. In general, an annulus is a ring-shaped region defined as the set of points in the plane that are between two concentric circles.

Bing's recognition theorem is a result in the field of topology, specifically in the study of 3-manifolds. It states that if a triangulated 3-manifold is homeomorphic to a simplicial complex, then it can be recognized topologically by its triangulation. In other words, the theorem provides conditions under which one can determine whether two triangulated 3-manifolds are homeomorphic based solely on their combinatorial or geometric properties.

"Bing shrinking" refers to a phenomenon where Microsoft's Bing search engine experiences a decline in its market share or usage compared to its competitors, particularly Google. This can happen due to factors such as user preference, changes in search algorithms, or improvements in competitors' services. The term may also pertain to specific features or services within Bing being scaled back or removed.

The Blaschke selection theorem is a result in complex analysis and functional analysis concerning the behavior of sequences of Blaschke products, which are a type of analytic function associated with a sequence of points in the unit disk in the complex plane.

The Borromean rings are a set of three interlinked rings that are arranged in such a way that no two rings are directly linked together; instead, all three are interlinked with one another as a complete set. The key property of the Borromean rings is that if any one of the rings is removed, the remaining two rings will be unlinked, meaning they will not be entangled with each other.

The term "boundary parallel" can refer to different concepts depending on the context in which it is used. Generally, it relates to the idea of being aligned or closely associated with the boundaries of a particular system, area, or set of parameters. 1. **In Mathematics and Geometry**: Boundary parallel could describe lines, planes, or surfaces that run parallel to the edges or boundaries of a geometric shape or figure.

Boy's surface is a non-orientable surface that is an example of a mathematical structure in topology. It is a kind of 2-dimensional manifold that cannot be embedded in three-dimensional Euclidean space without self-intersections. Specifically, it can be constructed as a quotient of the 2-dimensional disk, and it can be visualized as a specific kind of "twisted" surface.

A **branched surface** is a concept in topology that can be thought of as a surface that has branching structures or singular points, where the usual notion of a smooth manifold breaks down. More specifically, branched surfaces arise in the study of topology and geometric structures where traditional structures—such as smooth surfaces—may not be adequate to describe certain features or behaviors.

A Casson handle is a mathematical concept used in the field of 3-manifold topology, specifically in the study of 3-manifolds and their structures. To understand what a Casson handle is, it's essential to first understand its role in manifolds and handle decompositions. In topology, a *handle* is a basic building block used to construct manifolds.

The Casson invariant is an important concept in the field of 3-manifold topology, particularly in relation to the study of oriented homology 3-spheres. It is a topological invariant associated with a 3-manifold that provides a measure of the manifold's structure, particularly focusing on the presence of certain types of surfaces and knots within the manifold.

Cellular decomposition is a concept in mathematics, particularly in topology and algebraic topology, that refers to the process of breaking down a topological space into simpler, more manageable pieces called cells. Cells are basic building blocks that can be thought of as generalizations of simple geometric shapes like points, line segments, disks, or higher-dimensional analogs.