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Lecture# Homology Theorem

Description

This lecture covers the proof of Theorem A, focusing on an example related to simplicial and singular homology, the Mayer-Vietoris sequence, naturality, and excision. The theorem states that for a good pair (X,A), the quotient map induces isomorphisms between homology groups. It also discusses the concept of deformation retracts and open neighborhoods. The lecture emphasizes the importance of understanding homotopy equivalences and the commutative diagrams involved in the proof.

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In course

MATH-323: Algebraic topology

Homology is one of the most important tools to study topological spaces and it plays an important role in many fields of mathematics. The aim of this course is to introduce this notion, understand it

Related concepts (65)

Simplicial homology

In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex).

Simplicial complex

In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex.

Simplicial set

In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.

Christians

Christians (ˈkɹɪstʃən,_-tiən) are people who follow or adhere to Christianity, a monotheistic Abrahamic religion based on the life and teachings of Jesus Christ. The words Christ and Christian derive from the Koine Greek title Christós (Χριστός), a translation of the Biblical Hebrew term mashiach (מָשִׁיחַ) (usually rendered as messiah in English). While there are diverse interpretations of Christianity which sometimes conflict, they are united in believing that Jesus has a unique significance.

Homology sphere

In algebraic topology, a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer . That is, and for all other i. Therefore X is a connected space, with one non-zero higher Betti number, namely, . It does not follow that X is simply connected, only that its fundamental group is perfect (see Hurewicz theorem). A rational homology sphere is defined similarly but using homology with rational coefficients.

Related lectures (13)

Covers the bar construction method, homology groups, classifying space, and the Hopf formula.

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Explores the topological Künneth Theorem, emphasizing commutativity and homotopy equivalence in chain complexes.

Demonstrates the equivalence between simplicial and singular homology, proving isomorphisms for finite s-complexes and discussing long exact sequences.

Covers the first properties of singular homology and the preservation of decomposition and path-connected components in topological spaces.