Homology theory

Homology theory is a branch of algebraic topology that studies topological spaces through the use of algebraic structures, primarily by associating a sequence of abelian groups or modules, called homology groups, to a topological space. These groups encapsulate information about the space's shape, connectivity, and higher-dimensional features.

Borel–Moore homology is a homological algebraic concept that arises in the study of algebraic varieties, particularly in the context of algebraic geometry and algebraic topology. It is a form of homology theory designed to handle locally compact topological spaces, with particular application to appropriate classes of varieties, such as quasi-projective varieties or complex algebraic varieties.

"Bump and hole" is a term that can have different meanings depending on the context, but it is often associated with construction, civil engineering, or road maintenance, referring to an issue related to road surfaces. When roads develop bumps and holes (or potholes), it can lead to uneven driving surfaces that can be dangerous for vehicles and pedestrians. In a more technical sense, "bump" refers to elevated areas on a surface, while "hole" refers to depressions.

Cellular homology is a tool in algebraic topology that allows for the computation of homology groups of a topological space by using a cellular structure derived from a CW-complex. A CW-complex is a kind of topological space that is built up from basic building blocks called cells, which are homeomorphic to open disks in Euclidean space, glued together in a specific way.

In algebraic topology, the cohomology ring is an important algebraic structure associated with a topological space. It is formed from the cohomology groups of the space, which provide algebraic invariants that help in understanding the topological properties of spaces.

Compactly supported homology is a version of homology theory that focuses on the study of spaces where the singular chains are required to have compact support. This concept is particularly useful in various areas of mathematics, including algebraic topology and differential geometry. ### Key Concepts: 1. **Homology**: Homology is a tool used in algebraic topology to study topological spaces by associating sequences of abelian groups (or modules) to them.

In mathematics, the term "continuation map" can refer to different concepts depending on the context, particularly in the realms of topology, functional analysis, and other areas of mathematics related to the study of continuous mappings and their properties. Here are a few interpretations: 1. **Topological Continuation Map**: In topology, a continuation map may refer to a function that extends (or continues) a function defined on a smaller space to a larger space while preserving certain properties, like continuity.

In category theory, a **cyclic category** typically refers to a category that captures the idea of cycles or circular structures. It can be viewed as a specialized type of category that includes objects and morphisms that relate to cyclical processes or relationships.

The Eilenberg-Moore spectral sequence is a mathematical construct used in the field of algebraic topology and homological algebra. It arises in the context of homotopical algebra, particularly when dealing with fibred categories and the associated homotopy theoretic situations.

The Eilenberg–Steenrod axioms are a set of axioms in algebraic topology that characterize (reduced) singular homology and cohomology theories. Formulated by Samuel Eilenberg and Norman Steenrod in the mid-20th century, these axioms provide a rigorous framework for what constitutes a generalized homology or cohomology theory. They serve as a foundation for the study of topological spaces through algebraic means.

The Excision Theorem is a fundamental result in algebraic topology, particularly in the context of singular homology. It addresses how the homology groups of a topological space can be affected by the removal of a "nice" subspace.

Graph homology is a concept in algebraic topology that extends the ideas of homology from topological spaces to combinatorial structures known as graphs. Essentially, it assigns algebraic invariants to graphs that capture their topological properties, allowing one to study and classify graphs in a way that is analogous to how homology groups classify topological spaces. ### Key Elements of Graph Homology 1. **Graphs**: A graph consists of vertices and edges connecting pairs of vertices.

The Hodge conjecture is a fundamental statement in algebraic geometry and topology that relates the topology of a non-singular projective algebraic manifold to its algebraic cycles. Formulated by W.V. Hodge in the mid-20th century, the conjecture suggests that certain classes of cohomology groups of a projective algebraic variety have a specific geometric interpretation.

Homological connectivity is a concept from algebraic topology and homological algebra that relates to how well-connected a topological space or algebraic object is in terms of its homological properties. It can involve examining the relationships between different homology groups of a space. In a more specific context, homological connectivity can refer to the lowest dimension in which the homology groups of a space are nontrivial.

Homology is a concept in mathematics, specifically in algebraic topology, that provides a way to associate a sequence of algebraic structures, such as groups or rings, to a topological space. This construction helps to analyze the shape or structure of the space in a more manageable form.

A homology sphere is a topological space that behaves like a sphere in terms of its homological properties, even if it is not actually a sphere in the classical sense. More formally, an \( n \)-dimensional homology sphere is a manifold that is homotopy equivalent to the \( n \)-dimensional sphere \( S^n \), and, importantly, it has the same homology groups as \( S^n \).

Hurewicz's theorem is a result in algebraic topology that pertains to the relationship between the homology and homotopy groups of a space. It specifically addresses the connection between the homology of a space and its fundamental group, particularly for spaces with certain properties.

K-homology is a cohomology theory in the field of algebraic topology that provides a way to study topological spaces using tools from K-theory. It is a variant of K-theory where one considers the behavior of vector bundles and their generalizations over spaces. K-homology is mainly applied in the framework of noncommutative geometry and has connections to several areas such as differential geometry, the theory of operator algebras, and index theory.

The Kan-Thurston theorem is a result in the field of topology and geometric group theory, particularly concerning the relationships between 3-manifolds and the algebraic properties of groups. More specifically, it is related to the conjecture regarding the recognition of certain types of 3-manifolds and the structures of groups that can be associated with them.

Khovanov homology is a mathematical invariant associated with knots and links in three-dimensional space. It was introduced by Mikhail Khovanov in 1999 as a categorification of the Jones polynomial, which is a well-known knot invariant.

The Kirby–Siebenmann class is a concept in the field of algebraic topology, particularly within the study of manifolds and their embeddings. It is named after mathematicians Robion Kirby and Louis Siebenmann, who introduced it in their work on the topology of high-dimensional manifolds. In particular, the Kirby–Siebenmann class arises in the context of the study of manifold structures and their differentiability.

The Mayer–Vietoris sequence is a fundamental tool in algebraic topology, particularly in the study of singular homology and cohomology theories. It provides a way to compute the homology or cohomology of a topological space from that of simpler subspaces.

Morse homology is a tool in differential topology and algebraic topology that studies the topology of a smooth manifold using the critical points of smooth functions defined on the manifold. It relates the topology of the manifold to the critical points of a Morse function, which is a smooth function where all critical points are non-degenerate (i.e., each critical point has a Hessian that is non-singular).

Poincaré duality is a fundamental theorem in algebraic topology that describes a duality relationship between certain topological spaces, particularly manifolds, and their cohomology groups. Named after the French mathematician Henri Poincaré, the theorem specifically applies to compact, oriented manifolds.

Polar homology is an algebraic concept that arises in the study of commutative algebra and algebraic geometry, particularly in the context of the theory of Gröbner bases and polynomial ideals. Polar homology can be thought of as a homology theory that is related to the structure of a polynomial ring, considering the "polar" aspects of a given polynomial or collection of polynomials.

The Pontryagin product is a way to define a multiplication operation on the cohomology ring of a topological group, specifically in the context of homotopy theory and algebraic topology. Named after the mathematician Lev Pontryagin, this product provides a rich algebraic structure that captures important information about the topological properties of the space.

In the context of algebraic topology, particularly in homology theory, the term "pushforward" refers to a specific kind of construction related to the behavior of homology classes under continuous maps between topological spaces.

Reduced homology is a variant of standard homology theory in algebraic topology, typically applied to topological spaces. It is particularly useful for spaces that are not simply connected or that have certain types of singularities, as it helps to simplify some aspects of their homological properties.

Relative Contact Homology (RCH) is a modern invariant in symplectic and contact topology, developed as a tool for studying contact manifolds. It serves as a means of categorifying certain notions from classical contact topology and provides insights into the geometry and topology of contact manifolds when compared to other invariants.

Relative homology is a concept in algebraic topology that extends the notion of homology groups to pairs of spaces. Specifically, if we have a topological space \( X \) and a subspace \( A \subseteq X \), the relative homology groups \( H_n(X, A) \) provide information about the structure of \( X \) relative to the subspace \( A \).

Simplicial volume is a notion from the field of topology and geometry, particularly in the study of manifolds. It essentially provides a way to measure the "size" of a topological manifold in a geometric sense. The concept is closely associated with the study of manifolds and their geometric structures, especially in the context of algebraic topology.

Singular homology is an important concept in algebraic topology, which provides a way to associate a sequence of abelian groups or vector spaces (called homology groups) to a topological space. These groups encapsulate information about the space's structure, such as its number of holes in various dimensions. ### Key Concepts: 1. **Simplices**: The building blocks of singular homology are simplices, which are generalizations of triangles.

The Steenrod problem, named after mathematician Norman Steenrod, refers to a question in the field of algebraic topology concerning the properties and structure of cohomology operations. Specifically, it deals with the problem of determining which cohomology operations can be represented by "natural" cohomology operations on spaces, particularly focusing on the stable homotopy category.

Stratifold is a computational tool used in the field of genomics and molecular biology to predict and analyze the folding structures of proteins. It applies algorithms rooted in statistical mechanics and machine learning to assess how proteins fold into their three-dimensional shapes based on their amino acid sequences. Understanding protein folding is crucial for deciphering biological functions and the development of pharmaceuticals, as misfolded proteins can lead to various diseases.

The Toda–Smith complex is a construction in algebraic topology, specifically in the study of spectra and homotopy theory. It is named after the mathematicians Hirosi Toda and Michael Smith, who contributed to the understanding of stable homotopy types and complex structures. More precisely, the Toda–Smith complex can be constructed from a simplicial set that illustrates certain relationships and equivalences in stable homotopy categories.