Induced homomorphism
Encyclopedia
In mathematics
, an induced homomorphism is a structure-preserving map between a pair of objects that is derived in a canonical way from another map between another pair of objects. A particularly important case arises in algebraic topology
, where any continuous function
between two pointed topological space
s induces a group homomorphism
between the fundamental group
s of the two spaces. Likewise, the same continuous map induces a group homomorphism between the respective homotopy group
s, the respective homology groups and a homomorphism going in the opposite direction between the corresponding cohomology groups.
A homomorphism
is a structure-preserving map between two mathematical objects of the same type: a group homomorphism
, for instance, is a map between two groups
such that the image of the product of any two group items is the same as the product of their images, while a graph homomorphism
is a map from the vertices of one undirected graph to the vertices of another such that any edge of the first graph is mapped to an edge of the second. Families of objects, and maps between them, are generally formalized as objects and morphisms in a category
; by convention, the morphisms in categories are depicted as arrows in diagrams. In many of the
important categories of mathematics, the morphisms are called homomorphisms. In category theory, a functor
is itself a structure-preserving map, between categories: it must map objects to objects, and morphisms to morphisms, in a way that is compatible with the composition of morphisms within the category. If F is a functor from category A to category B, ƒ is a morphism in category A, and the morphisms of category B are called homomorphisms, then F(ƒ) is the homomorphism induced from ƒ by F.
For example, let X and Y be topological spaces with fundamental groups π(X,x0) and π(Y,y0) respectively, with specified base points
x0 and y0. If ƒ is a continuous function from X to Y that maps the base points to each other (that is, ƒ(x0) = y0) then any loop
based at x0 may be composed with ƒ to make a loop based at y0. This map of loops respects homotopy equivalence of loops: one can map any element of π(X,x0) to π(Y,y0) by choosing a loop representing the element, using ƒ to map that representative loop to Y, and selecting the homotopy equivalence class of the resulting mapped loop. Thus, ƒ corresponds to a homomorphism of fundamental groups; this homomorphism is called the induced homomorphism of ƒ. The construction of a fundamental group for each topological space, and of an induced homomorphism of fundamental groups for each continuous function, forms a functor from the category of topological spaces to the category of groups. See fundamental group#Functoriality for more on this type of induced homomorphism.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an induced homomorphism is a structure-preserving map between a pair of objects that is derived in a canonical way from another map between another pair of objects. A particularly important case arises in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, where any continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
between two pointed topological space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
s induces a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
between the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
s of the two spaces. Likewise, the same continuous map induces a group homomorphism between the respective homotopy group
Homotopy group
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...
s, the respective homology groups and a homomorphism going in the opposite direction between the corresponding cohomology groups.
A homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
is a structure-preserving map between two mathematical objects of the same type: a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
, for instance, is a map between two groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
such that the image of the product of any two group items is the same as the product of their images, while a graph homomorphism
Graph homomorphism
In the mathematical field of graph theory a graph homomorphism is a mapping between two graphs that respects their structure. More concretely it maps adjacent vertices to adjacent vertices.-Definitions:...
is a map from the vertices of one undirected graph to the vertices of another such that any edge of the first graph is mapped to an edge of the second. Families of objects, and maps between them, are generally formalized as objects and morphisms in a category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
; by convention, the morphisms in categories are depicted as arrows in diagrams. In many of the
important categories of mathematics, the morphisms are called homomorphisms. In category theory, a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
is itself a structure-preserving map, between categories: it must map objects to objects, and morphisms to morphisms, in a way that is compatible with the composition of morphisms within the category. If F is a functor from category A to category B, ƒ is a morphism in category A, and the morphisms of category B are called homomorphisms, then F(ƒ) is the homomorphism induced from ƒ by F.
For example, let X and Y be topological spaces with fundamental groups π(X,x0) and π(Y,y0) respectively, with specified base points
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
x0 and y0. If ƒ is a continuous function from X to Y that maps the base points to each other (that is, ƒ(x0) = y0) then any loop
Loop group
In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. Specifically, letLG \,denote the space of continuous mapsS^1 \to G...
based at x0 may be composed with ƒ to make a loop based at y0. This map of loops respects homotopy equivalence of loops: one can map any element of π(X,x0) to π(Y,y0) by choosing a loop representing the element, using ƒ to map that representative loop to Y, and selecting the homotopy equivalence class of the resulting mapped loop. Thus, ƒ corresponds to a homomorphism of fundamental groups; this homomorphism is called the induced homomorphism of ƒ. The construction of a fundamental group for each topological space, and of an induced homomorphism of fundamental groups for each continuous function, forms a functor from the category of topological spaces to the category of groups. See fundamental group#Functoriality for more on this type of induced homomorphism.
See also
- Induced homomorphism (algebraic topology)Induced homomorphism (algebraic topology)In mathematics, especially in the area of topology known as algebraic topology, an induced homomorphism is a way of relating the algebraic invariants of topological spaces which are already related by a continuous function. Such homomorphism exist whenever the algebraic invariants are functorial...
- Induced homomorphism (fundamental group)Induced homomorphism (fundamental group)In mathematics, especially in the area of topology known as algebraic topology, the induced homomorphism is a group homomorphism related to the study of the fundamental group.-Definition:...