A glossary of algebraic topology includes definitions and explanations of key terms and concepts within the field. Here’s a selection of important terms: 1. **Algebraic Topology**: A branch of mathematics concerned with the study of topological spaces through algebraic methods. 2. **Topological Space**: A set of points, along with a set of neighborhoods for each point that satisfies certain axioms.

In algebraic topology, a **good cover** refers to a specific type of open cover for a topological space, often in the context of the study of sheaf theory and cohomology.

Gray's conjecture is a statement in the field of combinatorial geometry, specifically related to the geometry of polytopes and projections. Proposed by the mathematician John Gray in the late 20th century, it posits that for any configuration of points in a Euclidean space, there exists a certain number of projections and arrangements that satisfy specific geometric properties.

A *cosheaf* is a mathematical concept used in the field of sheaf theory, which is a branch of topology and algebraic geometry. In general, a sheaf assigns algebraic or topological data to open sets of a topological space in a consistent manner, allowing one to "glue" data from smaller sets to larger ones.

Homeotopy refers to a concept in topology, a branch of mathematics that deals with properties of space that are preserved under continuous transformations. Specifically, the term "homeotopy" is often used interchangeably with "homotopy," which describes a way of continuously transforming one continuous function into another.

In algebraic topology, the concept of the homotopy fiber is a key tool used to study maps between topological spaces. It can be considered as a generalization of the notion of the fiber in the context of fibration, and it helps to understand the homotopical properties of the map in question.

The term "local system" can refer to different concepts depending on the context in which it is used. Here are a few common interpretations: 1. **Local Area Network (LAN)**: In computing, a local system often refers to devices and computers connected within a limited geographical area, such as a home, office, or school. This can include computers, printers, and other devices that communicate with each other using a local network, often without accessing the broader internet.

A cyclic cover, in mathematics, is often associated with certain concepts in algebraic geometry and number theory, particularly in the study of covering spaces and families of algebraic curves. Here are some contexts in which the term "cyclic cover" might be used: 1. **Covering Spaces in Topology**: In topology, a cyclic cover refers to a specific type of covering space where the fundamental group of the base space acts transitively on the fibers of the cover.

Deligne's conjecture on Hochschild cohomology is a significant statement in the realm of algebraic geometry and homological algebra, particularly relating to the Hochschild cohomology of categories of coherent sheaves. Formulated by Pierre Deligne in the late 20th century, the conjecture concerns the relationship between the Hochschild cohomology of a smooth proper algebraic variety and the associated derived categories.

The term "Delta set" can refer to different concepts depending on the context in which it is used. Here are a few possible interpretations across various fields: 1. **Mathematics/Statistics**: In statistics, a "delta set" could refer to a set of differences or changes between two datasets. For example, if you are comparing the performance of a variable over two different time periods, the delta set might represent the changes observed.

Derived algebraic geometry is a modern field of mathematics that extends classical algebraic geometry by incorporating tools and concepts from homotopy theory, derived categories, and categorical methods. It aims to refine the geometric and algebraic structures used to study schemes (the fundamental objects of algebraic geometry) by considering them in a more flexible and nuanced framework that can handle various kinds of singularities and complex relationships.

A "generalized map" can refer to different concepts depending on the context in which it is used. Here are a few interpretations based on various fields: 1. **Mathematics/Topology**: In topology, a generalized map might refer to a continuous function that extends the idea of mapping beyond traditional functions. For example, in homotopy theory, generalized maps could involve mappings between topological spaces that account for more abstract constructs like homotopies or morphisms.

In the context of topology, a **join** is an operation that combines two topological spaces into a new space. Given two topological spaces \( X \) and \( Y \), the join of \( X \) and \( Y \), denoted \( X * Y \), is constructed in a specific way. The join \( X * Y \) can be visualized as follows: 1. **Take the Cartesian product** \( X \times Y \).

In topology, the symmetric product of a topological space \( X \), denoted as \( S^n(X) \), is a way to construct a new space from \( X \) that encodes information about \( n \)-tuples of points in \( X \) while factoring in the notion of indistinguishability of points.

Eckmann–Hilton duality is a concept in algebraic topology and category theory that describes a relationship between certain algebraic structures, particularly in the context of homotopy theory and higher algebra. It emerges in the study of operads and algebraic models of spaces, particularly homotopy types and their associated algebraic invariants. The duality is expressed within the framework of category theory, particularly in the context of monoidal categories and homotopy coherent diagrams.

An Eilenberg-MacLane spectrum is a fundamental concept in stable homotopy theory, and it is used to represent cohomology theories in the context of stable homotopy categories. Specifically, for an Abelian group \( G \), the Eilenberg-MacLane spectrum \( H\mathbb{Z}G \) can be thought of as a spectrum that represents the homology or cohomology theory associated with the group \( G \).

Topological graph theory is a branch of mathematics that studies the interplay between graph theory and topology. It focuses primarily on the properties of graphs that are invariant under continuous transformations, such as stretching or bending, but not tearing or gluing. The key aspects of topological graph theory include: 1. **Graph Embeddings**: Understanding how a graph can be drawn in various surfaces (like a plane, sphere, or torus) without edges crossing.

K-theory is a branch of mathematics that studies vector bundles and more generally, topological spaces and their associated algebraic invariants. It has applications in various fields, including algebraic geometry, operator theory, and mathematical physics. The core idea in K-theory involves the classification of vector bundles over a topological space. Specifically, there are two main types of K-theory: 1. **Topological K-theory**: This version studies topological spaces and their vector bundles.

Double torus knots and links are concepts from the field of knot theory, which is a branch of topology. In topology, knots are considered as embeddings of circles in three-dimensional space, and links are collections of such embeddings. ### Double Torus A double torus is a surface that is topologically equivalent to two tori (the plural of torus) connected together. It's often visualized as the shape of a "figure eight" or a surface with two "holes.