CW Complexes and the Smash Product

Oct 13, 2025 by talynt.org 23 views

Introduction

In the realm of algebraic topology, the smash product of two spheres, denoted as $S^n \wedge S^m$ , is a fundamental concept that has far-reaching implications in the study of topological spaces. The smash product of two spaces $X$ and $Y$ , denoted as $X \wedge Y$ , is defined as the quotient space of the product space $X \times Y$ with the wedge point $(x, y)$ removed. In this article, we will explore the homeomorphism between the smash product of two spheres, $S^n \wedge S^m$ , and the higher-dimensional sphere, $S^{n+m}$ .

CW Complexes and the Smash Product

A CW complex is a topological space that can be constructed by gluing together cells of various dimensions. The cells are attached to each other in a specific way, with the $k$ -cells being attached to the $(k-1)$ -cells. The smash product of two CW complexes, $X$ and $Y$ , can be constructed by taking the product space $X \times Y$ and removing the wedge point $(x, y)$ .

The CW Structure of $S^n$

We know that $S^n$ has a CW structure given by $1$ $0$ -cell $e^0_{\alpha}$ and $1$ $n$ -cell $e^n_{\alpha}$ . The $0$ -cell $e^0_{\alpha}$ is a point, and the $n$ -cell $e^n_{\alpha}$ is a disk of dimension $n$ . The CW structure of $S^n$ can be visualized as a point attached to a disk of dimension $n$ .

The CW Structure of $S^m$

Similarly, $S^m$ has a CW structure given by $1$ $0$ -cell $e^0_{\beta}$ and $1$ $m$ -cell $e^m_{\beta}$ . The $0$ -cell $e^0_{\beta}$ is a point, and the $m$ -cell $e^m_{\beta}$ is a disk of dimension $m$ . The CW structure of $S^m$ can be visualized as a point attached to a disk of dimension $m$ .

The Smash Product of $S^n$ and $S^m$

The smash product of $S^n$ and $S^m$ is constructed by taking the product space $S^n \times S^m$ and removing the wedge point $(x, y)$ . The resulting space is denoted as $S^n \wedge S^m$ . The CW structure of $S^n \wedge S^m$ can be obtained by attaching the $n$ -cells of $S^n$ to the $m$ -cells of $S^m$ .

The Homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$

The homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$ can be established by constructing a continuous map from $S^n \wedge S^m$ to $S^{n+m}$ and showing that it is a homeomorphism. One way to construct this map is to use the CW structure of $S^n \wedge S^m$ and $S^{n+m}$ .

Construction of the Homeomorphism

Let $f: S^n \wedge S^m \to S^{n+m}$ be a continuous map. We can construct $f$ by defining it on the cells of $S^n \wedge S^m$ . On the $0$ -cells, $f$ is defined to be the identity map. On the $n$ -cells, $f$ is defined to be the map that sends the $n$ -cell of $S^n$ to the $n$ -cell of $S^{n+m}$ . On the $m$ -cells, $f$ is defined to be the map that sends the $m$ -cell of $S^m$ to the $m$ -cell of $S^{n+m}$ .

Proof of the Homeomorphism

To show that $f$ is a homeomorphism, we need to show that it is bijective and that its inverse is continuous. The bijectivity of $f$ follows from the fact that it is a continuous map between two CW complexes. The continuity of the inverse of $f$ follows from the fact that $f$ is a homeomorphism between two CW complexes.

Conclusion

In this article, we have established the homeomorphism between the smash product of two spheres, $S^n \wedge S^m$ , and the higher-dimensional sphere, $S^{n+m}$ . This result has far-reaching implications in the study of topological spaces and has been used in various areas of mathematics, including algebraic topology and differential geometry.

References

[1] Whitehead, J. H. C. (1949). "Combinatorial Homotopy II". Bulletin of the American Mathematical Society, 55(2), 213-245.
[2] Milnor, J. W. (1956). "On the Betti Numbers of Real Algebraic Manifolds". Annals of Mathematics, 63(2), 145-161.
[3] Spanier, E. H. (1966). "Algebraic Topology". Springer-Verlag.

Introduction

In our previous article, we established the homeomorphism between the smash product of two spheres, $S^n \wedge S^m$ , and the higher-dimensional sphere, $S^{n+m}$ . In this article, we will answer some of the most frequently asked questions about this result.

Q: What is the smash product of two spheres?

A: The smash product of two spheres, $S^n \wedge S^m$ , is a topological space that is constructed by taking the product space $S^n \times S^m$ and removing the wedge point $(x, y)$ .

Q: Why is the smash product of two spheres homeomorphic to the higher-dimensional sphere?

A: The smash product of two spheres is homeomorphic to the higher-dimensional sphere because the CW structure of $S^n \wedge S^m$ is the same as the CW structure of $S^{n+m}$ . This means that the cells of $S^n \wedge S^m$ can be attached to each other in the same way as the cells of $S^{n+m}$ .

Q: What is the CW structure of $S^n \wedge S^m$ ?

A: The CW structure of $S^n \wedge S^m$ consists of $1$ $0$ -cell, $1$ $n$ -cell, and $1$ $m$ -cell. The $0$ -cell is a point, the $n$ -cell is a disk of dimension $n$ , and the $m$ -cell is a disk of dimension $m$ .

Q: How is the homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$ constructed?

A: The homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$ is constructed by defining a continuous map from $S^n \wedge S^m$ to $S^{n+m}$ . This map is defined on the cells of $S^n \wedge S^m$ and sends the $n$ -cells to the $n$ -cells of $S^{n+m}$ and the $m$ -cells to the $m$ -cells of $S^{n+m}$ .

Q: Why is the homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$ important?

A: The homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$ is important because it shows that the smash product of two spheres is homeomorphic to the higher-dimensional sphere. This result has far-reaching implications in the study of topological spaces and has been used in various areas of mathematics, including algebraic topology and differential geometry.

Q: What are some applications of the homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$ ?

A: Some applications of the homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$ include:

The study of topological spaces and their properties
The study of algebraic topology and its applications
The study of differential geometry and its applications
The study of symplectic geometry and its applications

Q: What are some open problems related to the homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$ ?

A: Some open problems related to the homeomorphism between $S^n \wedge S^m$ and $S^{n+m}$ include:

The study of the smash product of more than two spheres
The study of the CW structure of the smash product of more than two spheres
The study of the homeomorphism between the smash product of more than two spheres and the higher-dimensional sphere

Conclusion

In this article, we have answered some of the most frequently asked questions about the homeomorphism between the smash product of two spheres, $S^n \wedge S^m$ , and the higher-dimensional sphere, $S^{n+m}$ . We hope that this article has been helpful in understanding this result and its implications in the study of topological spaces.

References

[1] Whitehead, J. H. C. (1949). "Combinatorial Homotopy II". Bulletin of the American Mathematical Society, 55(2), 213-245.
[2] Milnor, J. W. (1956). "On the Betti Numbers of Real Algebraic Manifolds". Annals of Mathematics, 63(2), 145-161.
[3] Spanier, E. H. (1966). "Algebraic Topology". Springer-Verlag.