Relative Homology
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Relative Homology
In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for topological pair, pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace. Definition Given a subspace A\subseteq X, one may form the short exact sequence :0\to C_\bullet(A) \to C_\bullet(X)\to C_\bullet(X) /C_\bullet(A) \to 0 , where C_\bullet(X) denotes the singular chains on the space ''X''. The boundary map on C_\bullet(X) leaves C_\bullet(A) invariant and therefore descends to a boundary map \partial'_\bullet on the quotient. If we denote this quotient by C_n(X,A):=C_n(X)/C_n(A), we then have a complex :\cdots\longrightarrow C_n(X,A) \xrightarrow C_(X,A) \longrightarrow \cdots . By definition, the th relative homology group of the pair of spaces (X,A) is :H_n(X,A) := \ker\partial'_n/\operator ...
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Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gr ...
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