Cyclic homology

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.

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 parts of mathematics, such as de Rham cohomology, Hochschild homology, group cohomology, and K-theory. These links make cyclic homology a useful tool because it can bring ideas from different areas of mathematics together. Over the years, many mathematicians have helped develop this theory, including Max Karoubi, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, Jean-Louis Loday, Daniel Quillen, Joachim Cuntz, Ryszard Nest, Ralf Meyer, and Michael Puschnigg.

Hints about definition

Cyclic homology is a special way to study mathematical structures called rings. It was created using a method named the Connes complex, which builds on another method called the Hochschild homology complex. This helps mathematicians see important links between different parts of math.

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

Case of commutative rings

Cyclic cohomology for some types of algebras can be studied using a special math tool called the algebraic de Rham complex. When the algebra comes from smooth shapes, its cyclic cohomology links closely to de Rham cohomology. This helps mathematicians study similar ideas in more complex algebras.

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

Variants of cyclic homology

Cyclic homology is a simpler way to study K-theory, a complex math idea. It works better for special types of algebras, like Fréchet algebras or C*-algebras.

Mathematicians made new versions of cyclic homology to improve it for these algebras. These versions include entire cyclic homology by Alain Connes, analytic cyclic homology by Ralf Meyer, and asymptotic and local cyclic homology by Michael Puschnigg. These help link cyclic homology more closely to K-theory.

Applications

Cyclic homology helps mathematicians find new ways to understand important ideas, like the Atiyah-Singer index theorem. They use special methods called spectral triples and deformation quantization of Poisson structures.

Cyclic cohomology lets people study complex math problems with elliptic operators. It works for many kinds of spaces, including foliations, orbifolds, and singular spaces in noncommutative geometry.

Computations of algebraic K-theory

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

In 1986, a mathematician named Goodwillie discovered that for some types of math structures, this map can show that two different calculations give the same answer. Later work in 2018 expanded this idea, showing that these calculations match under many more conditions. This helps mathematicians solve hard problems by giving them new ways to look at them.