- #1
wofsy
- 726
- 0
I would like to know how to define the cup product in cubular homology theory.
jim mcnamara said:Do you mean the Massey product? Please define cubular.
morphism said:How do you define the cup product in simplicial homology? Did you mean cohomology? If so, then the cup product is defined pretty much the same way. Perhaps you can show us your definitions and explain what you're having trouble adapting.
morphism said:You still have that in cubical homology. In fact, you have the Eilenberg-Zilber theorem, which tells you that for any topological spaces X and Y there is a chain homotopy equivalence [itex]C_\ast(X\times Y) \to C_\ast(X)\otimes C_\ast(Y)[/itex].
morphism said:I don't really understand your question. The diagonal approximation is the same.
The diagonal map [itex]\Delta : X \to X \times X[/itex] induces a map [itex]C_\ast(X) \to C(X \times X)[/itex] which we can then compose with the Eilenberg-Zilber map given above to get a map [itex]C_\ast(X) \to C_\ast(X) \otimes C_\ast(X)[/itex].
If you want all the details, then check out chapter XIII in Massey's book "A Basic Course in Algebraic Topology" (Springer GTM127), where the cup product is defined. (Incidentally, Massey's book is the only place I've seen the cubical approach to singular (co)homology. Is Massey the guy who invented this?)
morphism said:Ah, I see what you're asking. You basically want the explicit Eilenberg-Zilber map for cubical homology. This isn't the most straightforward thing to write down, as you need a few intermediaries. But like I said, all the details are found in Massey. If you really need this stuff and can't get your hands on a copy, I can probably write it all out for you later.
Cubular homology cup product is a mathematical concept in algebraic topology that measures the interaction between two cycles in a topological space. It assigns a product to pairs of cycles, which is a higher-dimensional analog of the intersection of two curves on a surface.
The cubular homology cup product is calculated by defining a bilinear operation between two cycles, represented by simplicial chains, in a cubical complex. This operation is then extended to a product between two cochains, which can be used to compute the cup product of two cohomology classes.
The cubular homology cup product is an important tool in algebraic topology, as it allows for the study of algebraic structures on topological spaces. It is used to define more complicated algebraic structures, such as cohomology rings, which can provide information about the shape and geometry of a space.
Yes, cubular homology cup product has applications in various fields such as physics, computer science, and engineering. It can be used to study the topology of networks, analyze data, and understand the behavior of dynamical systems.
While cubular homology cup product is a powerful tool, it does have some limitations. It is only defined for certain types of spaces, such as cubical complexes, and may not be applicable to more general topological spaces. Additionally, computing the cup product can be computationally intensive, making it challenging to use in certain situations.