Retraction in surface of genus g

  • Thread starter ForMyThunder
  • Start date
  • Tags
    Surface
In summary, the conversation discusses a problem in algebraic topology regarding the non-retractability of a surface Mg onto a separating circle C. The solution considers the assumptions of a possible retraction and uses the fundamental group of C to show a contradiction, ultimately proving that Mg does not retract onto C.
  • #1
ForMyThunder
149
0

Homework Statement


In the surface Mg of genus g, let C be a circle that separates Mh' and Mk' obtained from the closed surfaces Mh and Mk by deleting an open disk from each. Show that Mh' does not retract onto its boundary circle C, and hence Mg does not retract onto C.

Hatcher Allen. Algebraic Topology Section 1.2 Problem 9


Homework Equations





The Attempt at a Solution

Suppose there was such a retraction. Then we would have that [tex]i_*:\pi_1(C)\to\pi_1(M'_h)[/tex] induced by the inclusion map is injective and that [tex]\phi:\pi_1(M'_h)\to\pi_1(M_h)[/tex] is surjective with kernel [itex]i_*(\pi_1(C))[/itex]. Thus, [tex]\pi_1(M'_h)/i_*(\pi_1(C))\cong \pi_1(M_h)[/tex] and by taking the abelianizations: [tex]\mathbb{Z}^{2h-1}\cong\mathbb{Z}^{2h}/\mathbb{Z}\cong\mathbb{Z}^{2h}[/tex] yielding a contradiction.

Is this correct? I used the assumption that C was a retract of Mh' to say that the fundamental group of C is isomorphic to a subgroup of the fundamental group of Mh'.
 
Physics news on Phys.org
  • #2
Actually, in that last line, [itex]\mathbb{Z}^{2h-1}\cong\mathbb{Z}^{2h}/\mathbb{Z}[/itex] is not necessarily true. But the second isomorphism in that line is implied by the previous line and this leads to the contradiction.
 

FAQ: Retraction in surface of genus g

1. What is the concept of retraction in surface of genus g?

Retraction in surface of genus g refers to the process of continuously shrinking a surface with g holes (or handles) until it becomes a point. This concept is often used in topology and geometry to study the properties of surfaces with different numbers of holes.

2. Why is retraction in surface of genus g important?

Retraction in surface of genus g is important because it allows us to understand the topological and geometric properties of surfaces with different numbers of holes. It helps us to classify and study these surfaces, and also has applications in various fields such as physics, engineering, and computer science.

3. How is retraction in surface of genus g different from retraction in other surfaces?

The concept of retraction is similar in all surfaces, but the difference lies in the number of holes or handles. In surfaces of genus g, there are g holes, while in other surfaces there may be a different number of holes or no holes at all. This difference affects the behavior and properties of the retraction process.

4. Can you give an example of retraction in surface of genus g?

One example of retraction in surface of genus g is the shrinking of a torus (a doughnut-shaped surface) with one hole until it becomes a point. During this process, the torus will go through various stages where it will have different numbers of holes, until it finally becomes a point. This process can also be visualized by imagining a rubber band being continuously stretched and pulled towards a point on the surface of the torus.

5. What are some real-world applications of retraction in surface of genus g?

Retraction in surface of genus g has various real-world applications, such as in physics where it is used to study the properties of spacetime and black holes. It is also used in computer science for data compression and in engineering for designing and analyzing structures with holes, such as bridges and tunnels. Additionally, retraction in surface of genus g has applications in biology, where it is used to understand and model the shapes of biological organisms with different numbers of holes.

Similar threads

Replies
21
Views
8K
Replies
1
Views
2K
Replies
25
Views
3K
Replies
33
Views
8K
4
Replies
137
Views
17K
2
Replies
46
Views
6K
2
Replies
61
Views
11K
3
Replies
93
Views
12K
2
Replies
52
Views
10K
Back
Top