A presheaf with transfers is a concept in the realm of algebraic geometry and homotopy theory, specifically in the study of sheaves and cohomological constructs. The notion is related to the idea of "transfers," which are maps that allow for the extension of certain algebraic structures across various bases or schemes.
A homotopy Lie algebra is an algebraic structure that arises in the context of homotopy theory, particularly in the study of spaces, their algebraic invariants, and the relationships between them. It generalizes the notion of a Lie algebra by allowing for "higher" homotopical information. ### Definition 1.
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.
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 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.
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.
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.
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 \( A \).
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.