String theory - Homology and Homotopy

[wodhas]

Hi,

I am trying to understand relations between Homology/CoHomology and Homotopy group..I am familiar with Hurewicz Theorem but I need something more general.

I hope that you can help.

Thanks,
Lala

 

[Haelfix]

I didn't understand the physicists treatment of the subject at all when I was first learning about it as an undergrad. So instead I took an algebraic geometry class in the math department, and well things became a lot clearer.

We used Alan Hatchers book (which you can find online at his Cornell site). Be warned, its quite opaque and difficult, but then once you get it, the physics stuff is relatively trivial by comparison and it has a pretty good review off the particular question you are asking.

If that's too much, then typically physicists learn their algebraic geometry from something like Nakahara, but then it won't get into the nitty gritty details of stuff like the Hurewicz theorem.

 

[sir-pinski]

Hi Lala,

String theory is a theoretical framework in physics that attempts to reconcile the two major theories of physics - general relativity and quantum mechanics. It proposes that the fundamental building blocks of the universe are not particles, but rather tiny strings vibrating at different frequencies. This theory has been developed to explain the fundamental forces of nature and the nature of matter.

In relation to Homology and Homotopy, these are mathematical concepts that are used to study the properties of spaces and the continuous transformations between them. Homology and cohomology are algebraic invariants that capture the global structure of a space, while homotopy groups are algebraic invariants that capture the local structure of a space.

The Hurewicz Theorem is a fundamental result in algebraic topology that relates homotopy groups to homology groups. It states that for a connected topological space, the first non-trivial homotopy group is isomorphic to the first homology group. This provides a way to compute homotopy groups using homology groups, which can be easier to calculate.

However, if you are looking for a more general understanding of the relationship between homology and homotopy, you may want to look into the concept of a homotopy type. A space is said to have the same homotopy type as another space if they are homotopy equivalent, meaning that they can be continuously deformed into each other. In this case, their homotopy groups and homology groups will also be isomorphic.

I hope this helps in your understanding of the connections between string theory, homology, and homotopy. Keep exploring and learning!