What is the fundamental group of the torus?
The fundamental group of an n-torus is a free abelian group of rank n. The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n.
How do you find the fundamental group?
Van Kampen’s theorem can be used to compute the fundamental group of a space in terms of simpler spaces it is constructed from. If certain conditions are met, the theorem states that for X=⋃Aα, π1(X)=∗απ1(Aα), the free product of the component fundamental groups.
What is the fundamental group of a torus with one point removed?
A torus with one point removed deformation retracts onto a figure eight, namely the union of two generating circles. More generally, a surface of genus g with one point removed deformation retracts onto a rose with 2g petals, namely the boundary of a fundamental polygon.
What is fundamental group give example?
Loosely speaking, the fundamental group measures “the number of holes” in a space. For example, the fundamental group of a point or a line or a plane is trivial, while the fundamental group of a circle is Z.
What is a 2D torus?
2D Torus: it is two dimension with degree of 4, the nodes are imagined laid out in a two-dimensional rectangular lattice of n rows and n columns, with each node connected to its 4 nearest neighbors, and corresponding nodes on opposite edges connected.
What are the two most fundamental groups of cells called?
There are only two main types of cells: prokaryotic and eukaryotic. Prokaryotic cells lack a nucleus and other membrane-bound organelles. Eukaryotic cells have a nucleus and other membrane-bound organelles. This allows these cells to have complex functions.
Is the Hawaiian earring compact?
The Hawaiian earring E is the compact subset of the xy-plane that is the union of the countably many circles Cn, where Cn has radius 1/n and center (0,−1/n). The earring is obviously both locally and globally path connected.
Is fundamental group Abelian?
The fundamental group is abelian iff basepoint-change homomorphisms depend only on the endpoints.
How many dimensions is a torus?
The torus itself is a 2-dimensional creature. If you want to embed it in Euclidean space, it is most natural to do so in 4-dimensional space, although it’s possible to embed it in 3-dimensional space as well.
Is every group a fundamental group?
Every group can be realized as the fundamental group of a connected CW-complex of dimension 2 (or higher). As noted above, though, only free groups can occur as fundamental groups of 1-dimensional CW-complexes (that is, graphs).
Is the Hawaiian earring a CW complex?
The Hawaiian earring is an example of a topological space that does not have the homotopy-type of a CW complex.
What is a double torus?
A double torus is a topological surface with two holes, formed from the connected sum of two tori. It has an orientable genus of 2 and an Euler characteristic of -2.
What does CW in CW complex stand for?
closure-finite
The C stands for “closure-finite”, and the W for “weak” topology. A CW complex can be defined inductively. A 0-dimensional CW complex is just a set of zero or more discrete points (with the discrete topology).
Are manifolds CW complexes?
Every compact smooth manifold admits a smooth triangulation and hence a CW-complex structure.
What is the equation of a torus?
Torus parametrization z = R2 sin(v) where u in [0, 2 Pi) is the angle about the z axis and v is in [0, 2 Pi). ( R1 – (x2 + y2)1/2 )2 + z2 = R22 The aspect ratio of the torus is R1 / R2.
Is a CW-complex a manifold?
Any smooth manifold admits a CW-structure. In fact it is known that any smooth manifold can be triangulated, and hence admits the structure of a simplicial complex (see example 2).
What is the fundamental group of Figure 8?
Undoubtedly you know that the fundamental group of figure 8 is that free product of two infinite cyclic groups. Your space is the direct product 8 × S 1. Show activity on this post.
How to compute the fundamental group of the space from two tori?
Compute the fundamental group of the space obtained from two tori ✕ S 1 ✕ S 1 by identifying a circle ✕ S 1 ✕ { x 0 } in one torus with the corresponding circle ✕ S 1 ✕ { x 0 } in the other.
Is γ a subgroup of the center of G?
Since G is a connected group with a continuous action by conjugation on a discrete group Γ, it must act trivially, so that Γ has to be a subgroup of the center of G. In particular π 1 ( H) = Γ is an abelian group; this can also easily be seen directly without using covering spaces. The group G is called the universal covering group of H .
What is the fundamental group of a bouquet of R circles?
More generally, the fundamental group of a bouquet of r circles is the free group on r letters. The fundamental group of a wedge sum of two path connected spaces X and Y can be computed as the free product of the individual fundamental groups: