# Homology and Homotopy groups from properties

• heras1985
In summary, the conversation discusses the search for results that provide homology and homotopy groups based on certain properties of a space, including examples such as contractible spaces and Eilenberg MacLane spaces. The speaker also asks for recommendations for books or resources that contain similar results. The conversation also mentions the use of techniques such as cellular homology and Seifert-van-Kampen and Mayer-Vietoris. It concludes with a suggestion to compute the homology of a closed surface as a good exercise.
heras1985
I am looking for results which provides the homology and homotopy groups from some property of the space.
For instance, if a space $$X$$ is contractible then $$H_0(X)=\mathbb{Z}$$ and $$H_n(X)=0$$ if $$n\neq 0$$. Another example is the Eilenberg MacLane spaces $$K(\pi,n)$$ where $$\pi_n(K(\pi,n))=\pi$$ and $$\pi_r(K(\pi,n))=0$$ if $$n\neq r$$. It is also known the result for the homology groups of the spheres.
Do you know some similar result or some book where I can find them?

Last edited:
quasar987 said:
I'm sure there are hundreds of such rules.

Here's one: http://en.wikipedia.org/wiki/Fundamental_class

Yeah, I'm sure that there are hundreds of such rules, but it is difficult to find these rules explicitely.

Have you heard of cellular homology? Readily let's you compute the homology groups of any CW complex.

Other than that, Seifert-van-Kampen and Mayer-Vietoris are your friend.

heras1985 said:
I am looking for results which provides the homology and homotopy groups from some property of the space.
For instance, if a space $$X$$ is contractible then $$H_0(X)=\mathbb{Z}$$ and $$H_n(X)=0$$ if $$n\neq 0$$. Another example is the Eilenberg MacLane spaces $$K(\pi,n)$$ where $$\pi_n(K(\pi,n))=\pi$$ and $$\pi_r(K(\pi,n))=0$$ if $$n\neq r$$. It is also known the result for the homology groups of the spheres.
Do you know some similar result or some book where I can find them?

In a good first book on algebraic topology you will find many homology computations.
Homotopy is much harder. Rational homotopy of simply connected spaces can be computed from minimal models of rational cohomology. This is a powerful technique.

A good exercise is to compute the homology of an arbitrary closed surface.

## 1. What is the definition of homology groups?

Homology groups are algebraic structures used to measure the topological properties of spaces. They are defined as the quotient groups of the free abelian groups generated by the chains of a given space, modulo the subgroup of boundaries.

## 2. How are homology groups and homotopy groups related?

Homotopy groups also measure topological properties of spaces, but focus on the connectedness of these spaces. Homology groups and homotopy groups are related through the concept of homology, which is a map between two spaces that preserves the algebraic structure of their homology groups.

## 3. What are some applications of homology and homotopy groups?

Homology and homotopy groups have various applications in mathematics, physics, and engineering. They are used in fields such as topology, algebraic geometry, and computational fluid dynamics to study the properties of spaces and their transformations.

## 4. Can homology and homotopy groups be calculated for any space?

Homology and homotopy groups can be calculated for any topological space. However, computing these groups can be challenging for complex spaces and may require advanced mathematical techniques.

## 5. What is the significance of higher homotopy and homology groups?

Higher homotopy and homology groups provide more detailed information about the structure of spaces. They can detect finer topological features, such as holes and higher-dimensional voids, that may not be captured by the lower groups. They are also useful in distinguishing between spaces that have the same lower homotopy and homology groups.

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