Cyclic homology

Cyclic homology is an important idea in advanced mathematics, especially in areas known as noncommutative geometry. It helps mathematicians study certain types of algebraic structures called associative algebras, which are like generalized versions of the way numbers work together. Cyclic homology was introduced in the 1980s by two mathematicians, Boris Tsygan for homology and Alain Connes for cohomology.

This concept connects to many other older parts of mathematics, such as de Rham cohomology, Hochschild homology, group cohomology, and K-theory. These links make cyclic homology a valuable tool because it can bring insights from different mathematical fields together. Over the years, many mathematicians, including Max Karoubi, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, Jean-Louis Loday, Daniel Quillen, Joachim Cuntz, Ryszard Nest, Ralf Meyer, and Michael Puschnigg, have contributed to developing this theory.

Hints about definition

Cyclic homology is a special way to study mathematical structures called rings. It was developed using a method called the Connes complex, which builds on another method called the Hochschild homology complex. This helps mathematicians understand deep connections between different areas of math.

Later, mathematicians found a more general way to think about cyclic homology using ideas called cyclic objects. This approach shows that cyclic homology is related to something called a derived functor. One important result is a sequence that links cyclic homology with Hochschild homology, called the periodicity sequence.

Case of commutative rings

Cyclic cohomology for certain types of algebras can be understood using a special mathematical tool called the algebraic de Rham complex. When the algebra comes from smooth geometric shapes, its cyclic cohomology connects closely to de Rham cohomology, which studies these shapes using calculus-style methods.

This relationship helps mathematicians explore similar ideas even for more complex, "noncommutative" algebras, a concept developed further by Alain Connes.

Main article: Grothendieck
Main articles: Algebraic de Rham complex, De Rham cohomology

Variants of cyclic homology

Cyclic homology was developed as a simpler way to study K-theory, which is a complex math idea. Unlike K-theory, cyclic homology works better when looking at special types of algebras, like Fréchet algebras or C*-algebras.

To improve cyclic homology for these special algebras, mathematicians created several versions. These include entire cyclic homology by Alain Connes, analytic cyclic homology by Ralf Meyer, and asymptotic and local cyclic homology by Michael Puschnigg. These new versions help connect cyclic homology more closely to K-theory.

Applications

Cyclic homology helps mathematicians find new ways to prove and expand important ideas, like the Atiyah-Singer index theorem. This includes using special methods called spectral triples and deformation quantization of Poisson structures.

Cyclic cohomology offers a way to study more complex math problems involving elliptic operators. It works not just for smooth shapes but also for more complicated spaces like foliations, orbifolds, and singular spaces found in noncommutative geometry.

Computations of algebraic K-theory

The cyclotomic trace map is a special tool that helps connect two big ideas in math: algebraic K-theory and cyclic homology. It acts like a bridge, letting us use one to understand the other.

A big discovery in 1986 by Goodwillie showed that for certain types of math structures, this map can actually prove that two different calculations give the same answer. Later work in 2018 expanded this idea even further, showing that these calculations match up under many more conditions. This helps mathematicians solve tough problems by giving them new ways to look at them.