Induced homomorphism (fundamental group)
Encyclopedia
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
.
h*(f 0 g) = h*(f(2t)) for t in [0,1/2] = (h*(f)) + (h*(g))
h*(f 0 g) = h*(g(2t-1)) for t in [1/2,1] = (h*(f)) + (h*(g))
so that h* is indeed a homomorphism.
is not homeomorphic to R2 for their fundamental groups are not isomorphic (their fundamental groups don’t have the same cardinality). A simply connected space cannot be homeomorphic to a non-simply connected space; one has a trivial fundamental group and the other does not.
2. Any two topological spaces have homomorphic fundamental groups (at a particular base point). See note 2 where h* is the homomorphism induced by the constant map. However, they need not have isomorphic fundamental groups (at a particular base point). This shows that the fundamental groups of any two topological spaces always have the same ‘group structure’.
3. The fundamental group of the unit circle is isomorphic to the group of integers. Therefore, the one-point compactification
of R has a fundamental group isomorphic to the group of integers (since the one-point compactification of R is homeomorphic to the unit circle). This also shows that the one-point compactification of a simply connected space need not be simply connected.
4. The converse of the theorem need not hold. For example, R2 and R3 have isomorphic fundamental groups but are still not homeomorphic. Their fundamental groups are isomorphic because each space is simply connected. However, the two spaces cannot be homeomorphic because deleting a point from R2 leaves a non-simply connected space but deleting a point from R3 leaves a simply connected space (If we delete a line lying in R3, the space wouldn’t be simply connected anymore. In fact this generalizes to Rn whereby deleting a (n − 2)-dimensional parallelepiped from Rn leaves a non-simply connected space).
5. If A is a strong deformation retract of a topological space X, then the inclusion map
from A to X yields an isomorphism between fundamental groups.
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...
, especially in the area of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
known as 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...
, the induced homomorphism is 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...
related to the study of 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...
.
Definition
Let X and Y be topological spaces; let x0 be a point of X and let y0 be a point of Y. If h is a continuous map from X to Y such that h(x0) = y0. Define a map h* from π1(X, x0) to π1(Y, y0) by composing a loop in π1(X, x0) with h to get a loop in π1(Y, y0). Then h* is a homomorphism between fundamental groups known as the homomorphism induced by h.- If ƒ is a loop in π1(X, x0), then h*(ƒ) is a loop in π1(Y, y0). It should be noted that h*(ƒ) is a continuous map from [0, 1] to Y , and h*(ƒ(0)) = h*(x0) = y0 and h*(ƒ(1)) = h*(x0) = y0.
- h is indeed a homomorphism. To avoid repetition, whenever &fnof is called; and g loops, they will be known as loops based at x0. If ƒ and g are two loops. Then 0 is the group operation on π1(X, x0) and + is the group operation on π1(Y, y0)]
h*(f 0 g) = h*(f(2t)) for t in [0,1/2] = (h*(f)) + (h*(g))
h*(f 0 g) = h*(g(2t-1)) for t in [1/2,1] = (h*(f)) + (h*(g))
so that h* is indeed a homomorphism.
- Checking h* is a function (i.e. every loop in π1(X, x0) gets mapped onto a unique loop in π1(Y, y0)) follows from the fact that if ƒ and g are loops in π1(X, x0) that are homotopic via the homotopy H, then h*(ƒ) and h*(g) are homotopic via the homotopy h*H.
Theorem
Suppose X and Y are two homeomorphic topological spaces. If h is a homeomorphism from X to Y, then the induced homomorphism, h* is an isomorphism between fundamental groups [where the fundamental groups are π1(X, x0) and π1(Y, y0) with h(x0) = y0]Proof
It has already been checked in note 2 that h* is a homomorphism. It remains to check that h* is bijective. If p is the inverse of h; then p* is the inverse of h*. This follows from the fact that (p(h))*(ƒ) = p*(h*(ƒ)) = ƒ = (h(p))*(ƒ) = h*(p*(ƒ)). If ƒ and g are two loops in X where ƒ is not homotopic to g, the h*(ƒ) is not homotopic to h*(g); if F is a homotopy between them, p*(F) would be a homotopy between ƒ and g. If k is any loop in π1(Y, y0) , then h*(p*(k)) = k where p*(k) is a loop in X. This shows that h* is bijective.Applications of the theorem
1. The torusTorus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
is not homeomorphic to R2 for their fundamental groups are not isomorphic (their fundamental groups don’t have the same cardinality). A simply connected space cannot be homeomorphic to a non-simply connected space; one has a trivial fundamental group and the other does not.
2. Any two topological spaces have homomorphic fundamental groups (at a particular base point). See note 2 where h* is the homomorphism induced by the constant map. However, they need not have isomorphic fundamental groups (at a particular base point). This shows that the fundamental groups of any two topological spaces always have the same ‘group structure’.
3. The fundamental group of the unit circle is isomorphic to the group of integers. Therefore, the one-point compactification
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of R has a fundamental group isomorphic to the group of integers (since the one-point compactification of R is homeomorphic to the unit circle). This also shows that the one-point compactification of a simply connected space need not be simply connected.
4. The converse of the theorem need not hold. For example, R2 and R3 have isomorphic fundamental groups but are still not homeomorphic. Their fundamental groups are isomorphic because each space is simply connected. However, the two spaces cannot be homeomorphic because deleting a point from R2 leaves a non-simply connected space but deleting a point from R3 leaves a simply connected space (If we delete a line lying in R3, the space wouldn’t be simply connected anymore. In fact this generalizes to Rn whereby deleting a (n − 2)-dimensional parallelepiped from Rn leaves a non-simply connected space).
5. If A is a strong deformation retract of a topological space X, then the inclusion map
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
from A to X yields an isomorphism between fundamental groups.
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 (Category theory)