Pushforward (cohomology)
Imagine you have a toy car and you are pushing it forward on the ground. When you push the car forward, it moves along a certain path. In math, we have a similar concept called "pushforward."
In a mathematical theory called cohomology, we can think of shapes or spaces as if they were made up of little toy cars. These toy cars are called "manifolds" or "spaces."
Now, let's say we have two different spaces, let's call them "Space A" and "Space B." And we have a special kind of map or function that takes points from Space A and puts them into Space B. This map is called the "pushforward."
When we apply the pushforward to our toy cars on Space A, it moves each car to a corresponding point on Space B. So, if we push a car forward in Space A, the pushforward will make it move to a new spot on Space B.
But here's the catch - some cars might get squished or stretched during the pushforward. Imagine if you had a little toy car that was made of rubber, and when you pushed it forward, it got stretched or squished. Similarly, the pushforward of a space can change the shape of the space. It can stretch or squeeze it.
In math, we use the pushforward concept in cohomology to study how spaces change when we apply a pushforward map to them. We can compare the shapes of the original space and the pushed forward space to see how they are different.
By studying these changes, we can learn a lot of interesting things about the spaces we are working with. We can understand their structure, properties, and how they relate to each other.
So, the pushforward in cohomology is like the action of pushing a toy car forward, but instead of physical toys, we use mathematical spaces. The pushforward moves points from one space to another and can change the shape of the space, helping us explore and understand different mathematical objects.
In a mathematical theory called cohomology, we can think of shapes or spaces as if they were made up of little toy cars. These toy cars are called "manifolds" or "spaces."
Now, let's say we have two different spaces, let's call them "Space A" and "Space B." And we have a special kind of map or function that takes points from Space A and puts them into Space B. This map is called the "pushforward."
When we apply the pushforward to our toy cars on Space A, it moves each car to a corresponding point on Space B. So, if we push a car forward in Space A, the pushforward will make it move to a new spot on Space B.
But here's the catch - some cars might get squished or stretched during the pushforward. Imagine if you had a little toy car that was made of rubber, and when you pushed it forward, it got stretched or squished. Similarly, the pushforward of a space can change the shape of the space. It can stretch or squeeze it.
In math, we use the pushforward concept in cohomology to study how spaces change when we apply a pushforward map to them. We can compare the shapes of the original space and the pushed forward space to see how they are different.
By studying these changes, we can learn a lot of interesting things about the spaces we are working with. We can understand their structure, properties, and how they relate to each other.
So, the pushforward in cohomology is like the action of pushing a toy car forward, but instead of physical toys, we use mathematical spaces. The pushforward moves points from one space to another and can change the shape of the space, helping us explore and understand different mathematical objects.