Cone in topological space Homotopy problem

In summary, the conversation discusses the concept of homotopy in relation to continuous functions in a topological space, specifically in the cone of a given space. The homework problem asks to show that any two continuous functions in this cone are homotopic, and also to find the fundamental group of the cone. The student is seeking hints or help in solving the problem, and the expert suggests considering a special function and using the composition of homotopies. Additionally, part (b) is a simple result of part (a).
  • #1
JoeSabs
9
0

Homework Statement



Let Y be a topological space. Let CY denote the cone on Y.

(a) Show that any 2 continuous functions f, g : X --> CY are homotopic.
(b) Find (pi)1 (CY, p).

Homework Equations



I have no idea. The professor said one problem would be way out in left, to see who could make the connections. I can't. lol

The Attempt at a Solution



See 2.!

I know I haven't made an attempt, so I'm not asking for an answer. Any hints or help is much appreciated.
 
Last edited:
Physics news on Phys.org
  • #2
Remember that if two functions [tex]f, g[/tex] are each homotopic to another function [tex]k[/tex], then you can compose the homotopies to see that [tex]f[/tex] is homotopic to [tex]g[/tex]. Keeping that in mind, can you find a function [tex]k: X \to CY[/tex] which is special somehow, and useful for making homotopies in this way?

Part (b) is an easy corollary of part (a).
 

FAQ: Cone in topological space Homotopy problem

1. What is the cone in topological space homotopy problem?

The cone in topological space homotopy problem is a mathematical problem that involves studying the properties of a cone-like space in relation to continuous deformations, or homotopies, of the space. This problem is important in topology, which is a branch of mathematics that studies the properties of geometric objects that remain unchanged under continuous deformations.

2. Why is the cone in topological space homotopy problem important?

The cone in topological space homotopy problem is important because it helps us understand the structure and behavior of topological spaces, which are used to model real-world objects and phenomena. In addition, solutions to this problem can be applied to other areas of mathematics, such as algebraic topology and differential geometry.

3. How is the cone in topological space homotopy problem solved?

The cone in topological space homotopy problem is typically solved using techniques from algebraic topology, such as homotopy groups and homology groups. These tools allow mathematicians to classify and compare different topological spaces based on their properties, and ultimately solve the problem.

4. What are some applications of the cone in topological space homotopy problem?

The cone in topological space homotopy problem has many applications in mathematics and other fields. In topology, it is used to study manifolds, which are higher-dimensional versions of surfaces, and to classify spaces based on their homotopy type. In physics, it is used to model the behavior of physical systems, such as fluid flows and electromagnetic fields. It also has applications in computer science, where it is used to analyze and design efficient algorithms.

5. Are there any open problems related to the cone in topological space homotopy problem?

Yes, there are still many open problems related to the cone in topological space homotopy problem. Some of these include finding new techniques to solve the problem, generalizing the problem to higher dimensions, and extending the problem to other areas of mathematics, such as algebraic geometry and differential topology. These open problems provide opportunities for further research and advancement in the field of topology.

Similar threads

T_{0} - Space Equivalent Definition
Replies
2
Views
984
Center of mass of spherical shell inside of cone (Apostol Problem).
Replies
3
Views
1K
Is It Possible to Invert a Homotopy?
Replies
5
Views
672
Optimization problem - right circular cylinder inscribed in cone
Replies
9
Views
3K
Path-connectedness for finite topological spaces
Replies
3
Views
2K
Finding homeomorphism between topological spaces
Replies
11
Views
2K
Is 1 in the Closure of (2,3] in the Standard Topology on the Real Numbers?
Replies
3
Views
1K
MHB Can a Topological Space Be Homotopy Equivalent to a Cylinder?
Replies
3
Views
2K
Complex numbers sequences/C is a metric space
Replies
15
Views
2K
Is Every Connected Metric Space Compact?
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top