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.
What is the fundamental group of the Klein bottle?
The fundamental group of the Klein bottle can be determined as the group of deck transformations of the universal cover and has the presentation ⟨a, b | ab = b−1a⟩.
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.
Are humans a torus?
And so if you deform the human body and its inner (GI tract) and outer (skin) surfaces into the simplest possible shape, you end up with a doughnut-shaped object, a torus. All the other openings into the body that aren’t part of the GI tract aren’t holes, topologically/mathematically speaking, they’re cavities.
Is a torus orientable?
Orientable surfaces are surfaces for which we can define ‘clockwise’ consistently: thus, the cylinder, sphere and torus are orientable surfaces. In fact, any two-sided surface in space is orientable: thus the disc, cylinder, sphere and n-fold torus, all with or without holes, are orientable surfaces.
Are humans Klein bottles?
A Klein bottle is a three-dimensional version of a mobius strip. As humans only see in three dimensions, the fourth dimension must be inferred from a three-dimensional representation.
Is a circle homotopy equivalent to a point?
Disclaimer: OBVIOUSLY a circle is NOT homotopic to a point. However, if we just pick one point x0 from S1, then the mapping cylinder looks just like a cone, with F(x,0)=x, F(x0,t)=x0 (which is a line going down the cone from the circle to the bottom apex), and F(x,1)=x0.
Is a sphere topologically equivalent to a circle?
Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid.
How many topological holes does a human have?
For a normal human male, discounting pores, we have 9 holes.
How are humans donuts?
Is a torus one sided?
Among the closed surfaces the sphere and the torus are two-sided, while the Klein surface is one-sided. As examples of two-sided and one-sided situations one may cite imbeddings of the circle in the Möbius strip.
What is the genus of an object?
The genus is a topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it.