# How Do You Calculate the Homology of Spaces in Algebraic Topology?

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In summary, the conversation discussed various topics related to cohomology, including calculating the integral homology of a space, finding non-trivial products in certain cases, and the structure of homotopy classes between a finite CW-complex and the unit circle. It also touched on equivariant cohomology in relation to discrete groups and the conditions for its freeness.
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Hey guys, i am studying cohomology by hatcher's. Could anyone provide me some ideas on these problems? Thank you all!

Let f : S2n-1 -> Sn denote a continuous map. Let Xf = D2n union f Sn be the space obtained by attaching a 2n-dim cell to Sn using the map f.
i). Calculate the integral homology of Xf .
ii). Find an n and an f such that H*(Xf ;Z=2) has a non-trivial product. Justify it.

Let X be a path-connected finite CW-complex and see S1 as the complex numbers
of norm 1. Let [X; S1] denote the set of homotopy classes [f] of maps f : X -> S1.
i). Show that the set [X; S1] has the structure of a group induced by the multiplication
in S1.
ii). Show that [X; S1] is naturally isomorphic to H1(X; Z) as groups.( use Eilenberg-Maclane space)

Let G be a discrete group and X be a G-space. Recall that the equivariant cohomology rings of X are defined as H*G(X;Z) = H*(X x_G EG; Z). i). Show that for any X, the natural map H*(X;Z) -> H*G(X;Z) is injective. ii). Give a necessary condition for H*G(X;Z) to be a free abelian group. iii). For what groups G is H*G(X;Z) free?

## 1. What is Algebraic Topology?

Algebraic Topology is a branch of mathematics that studies the properties of topological spaces by using algebraic techniques. It deals with the study of continuous functions and their topological properties, such as connectedness, compactness, and continuity.

## 2. What are the main applications of Algebraic Topology?

Algebraic Topology has a wide range of applications in various fields, including physics, engineering, computer science, and data analysis. It is used to study and understand complex systems, such as networks and shapes, and to solve problems in data analysis, machine learning, and computer vision.

## 3. What are the main challenges in solving problems in Algebraic Topology?

The main challenge in solving problems in Algebraic Topology is the abstraction and complexity of the concepts involved. It requires a deep understanding of both algebra and topology, as well as the ability to apply these concepts to real-world problems. Additionally, the use of advanced mathematical tools and techniques may also pose a challenge for some researchers.

## 4. How is Algebraic Topology related to other branches of mathematics?

Algebraic Topology is closely related to other branches of mathematics, such as algebra, geometry, and analysis. It uses algebraic techniques to study topological spaces, which are geometric objects. It also has connections to differential geometry, homological algebra, and category theory.

## 5. What are some current research topics in Algebraic Topology?

Some current research topics in Algebraic Topology include the study of topological data analysis, persistent homology, and applications of homotopy theory in machine learning and computer vision. Other areas of interest include the study of algebraic structures in topology, such as group actions, and the development of new techniques for solving problems in Algebraic Topology.

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