Is the fundamental group a functor?
Therefore forming fundamental groups is not a functor on connected spaces.
What is a fundamental group in topology?
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space.
Is Abelianization a functor?
Abelianization as a functor On objects: It sends each group to the quotient group by its commutator subgroup. The corresponding natural transformation is the quotient map.
What is Abelianization of a group?
The Abelianization of a group is defined in the following equivalent ways: It is the quotient of the group by its commutator subgroup: in other words, it is the group . It is the quotient of by the relation . It is an Abelian group such that there exists a surjective homomorphism with the following property.
Is fundamental group Abelian?
The fundamental group is abelian iff basepoint-change homomorphisms depend only on the endpoints.
Is Möbius strip a 4D?
The Moebius strip is a two-dimensional object. Also putting two of them together in some sort of ways will result as a Klein bottle, which is truely a 4 dimensional object.
What is the number of components of a torus link?
A torus link arises if p and q are not coprime (in which case the number of components is gcd ( p, q )). A torus knot is trivial (equivalent to the unknot) if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot .
What is a torus knot in math?
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way.
What is the stretch factor of a torus knot?
Torus knots are the only knots whose knot groups have nontrivial center (which is infinite cyclic, generated by the element in the presentation above). The stretch factor of the ( p, q) torus knot, as a curve in Euclidean space, is Ω(min( p, q )), so torus knots have unbounded stretch factors.
What is the complement of a torus knot in the 3-sphere?
The complement of a torus knot in the 3-sphere is a Seifert-fibered manifold, fibred over the disc with two singular fibres. Let Y be the p -fold dunce cap with a disk removed from the interior, Z be the q -fold dunce cap with a disk removed its interior, and X be the quotient space obtained by identifying Y and Z along their boundary circle.