Universal Covering As A Sufficient Condition For Composition Of Covering Maps To Be A Covering Map

Introduction

In the realm of general topology and covering spaces, the concept of universal covering plays a pivotal role in understanding the properties of covering maps. A universal covering of a space $B$ is a covering space $\tilde X$ together with a covering map $\tilde p: \tilde X \rightarrow B$ such that for any other covering space $p: X \rightarrow B$, there exists a covering map $f: \tilde X \rightarrow X$ such that $p \circ f = \tilde p$. In this article, we will explore the sufficient condition for the composition of covering maps to be a covering map, given the existence of an universal covering.

Universal Covering

A universal covering of a space $B$ is a covering space $\tilde X$ together with a covering map $\tilde p: \tilde X \rightarrow B$ such that for any other covering space $p: X \rightarrow B$, there exists a covering map $f: \tilde X \rightarrow X$ such that $p \circ f = \tilde p$. This means that the universal covering $\tilde X$ is a "universal" space that covers all other covering spaces of $B$.

Composition of Covering Maps

Given two covering maps $p_1: X_1 \rightarrow B$ and $p_2: X_2 \rightarrow B$, we can form the composition $p_2 \circ p_1: X_1 \rightarrow X_2$. However, this composition may not necessarily be a covering map. In this article, we will show that if there exists an universal covering $\tilde p: \tilde X \rightarrow B$, then the composition of covering maps is a covering map.

Sufficient Condition

Suppose we have a base space $B$ for which there exists an universal covering $\tilde p: \tilde X \rightarrow B$. Let $p_1: X_1 \rightarrow B$ and $p_2: X_2 \rightarrow B$ be two covering maps. We want to show that the composition $p_2 \circ p_1: X_1 \rightarrow X_2$ is a covering map.

Step 1: Existence of a Covering Map

Since $\tilde p: \tilde X \rightarrow B$ is an universal covering, there exists a covering map $f: \tilde X \rightarrow X_1$ such that $p_1 \circ f = \tilde p$. Similarly, there exists a covering map $g: \tilde X \rightarrow X_2$ such that $p_2 \circ g = \tilde p$.

Step 2: Composition of Covering Maps

We can now form the composition $p_2 \circ p_1: X_1 \rightarrow X_2$. We want to show that this composition is a covering map.

Step 3: Proof of Sufficiency

To prove that the composition $p_2 \circ p_1: X_1 \rightarrow X_2$ is a covering map, we need to show that for any point $x \in X_1$, there exists a neighborhood $U$ of $x$ such that $(p_2 \circ p_1)(U)$ is evenly covered by $p_2 \circ p_1$.

Let $x \in X_1$. Since $f: \tilde X \rightarrow X_1$ is a covering map, there exists a neighborhood $V$ of $f^{-1}(x)$ such that $f(V)$ is evenly covered by $f$. Let $W = g(V)$. Then $W$ is a neighborhood of $g^{-1}(x)$ and $g(W)$ is evenly covered by $g$.

Step 4: Even Covering

Since $g(W)$ is evenly covered by $g$, there exists a neighborhood $U$ of $x$ such that $U = f^{-1}(p_1^{-1}(g(W)))$. Then $U$ is a neighborhood of $x$ and $(p_2 \circ p_1)(U) = p_2(p_1(U)) = p_2(g(W))$.

Step 5: Proof of Even Covering

To prove that $(p_2 \circ p_1)(U)$ is evenly covered by $p_2 \circ p_1$, we need to show that for any point $y \in (p_2 \circ p_1)(U)$, there exists a neighborhood $V$ of $y$ such that $(p_2 \circ p_1)(V)$ is evenly covered by $p_2 \circ p_1$.

Let $y \in (p_2 \circ p_1)(U)$. Then there exists a point $z \in U$ such that $y = (p_2 \circ p_1)(z)$. Since $U$ is evenly covered by $f$, there exists a neighborhood $V$ of $z$ such that $f(V)$ is evenly covered by $f$. Then $p_1(V)$ is a neighborhood of $y$ and $(p_2 \circ p_1)(p_1(V)) = p_2(p_1(V))$.

Step 6: Proof of Even Covering

To prove that $(p_2 \circ p_1)(p_1(V))$ is evenly covered by $p_2 \circ p_1$, we need to show that for any point $w \in (p_2 \circ p_1)(p_1(V))$, there exists a neighborhood $W$ of $w$ such that $(p_2 \circ p_1)(W)$ is evenly covered by $p_2 \circ p_1$.

Let $w \in (p_2 \circ p_1)(p_1(V))$. Then there exists a point $v \in V$ such that $w = (p_2 \circ p_1)(p_1(v))$. Since $p_1(V)$ is evenly covered by $p_1$, there exists a neighborhood $W$ of $v$ such that $p_1(W)$ is evenly covered by $p_1$. Then $(p_2 \circ p_1)(W)$ is a neighborhood of $w$ and $(p_2 \circ p_1)(W)$ is evenly covered by $p_2 \circ p_1$.

Conclusion

In this article, we have shown that if there exists an universal covering $\tilde p: \tilde X \rightarrow B$, then the composition of covering maps is a covering map. This result provides a sufficient condition for the composition of maps to be a covering map.

References

• [1] Hurewicz, W., and H. Wallman. Dimension Theory. Princeton University Press, 1941.
• [2] Massey, W. S. Algebraic Topology: An Introduction. Springer-Verlag, 1991.
• [3] Munkres, J. R. Topology. Prentice Hall, 2000.

Glossary

• Covering map: A continuous map $p: X \rightarrow B$ such that for any point $b \in B$, there exists a neighborhood $U$ of $b$ such that $p^{-1}(U)$ is evenly covered by $p$.
• Evenly covered: A set $U$ is said to be evenly covered by a map $p: X \rightarrow B$ if for any point $x \in U$, there exists a neighborhood $V$ of $x$ such that $p(V)$ is evenly covered by $p$.
• Universal covering: A covering space $\tilde X$ together with a covering map $\tilde p: \tilde X \rightarrow B$ such that for any other covering space $p: X \rightarrow B$, there exists a covering map $f: \tilde X \rightarrow X$ such that $p \circ f = \tilde p$.
  Universal Covering as a Sufficient Condition for Composition of Covering Maps to be a Covering Map: Q&A ===========================================================

Introduction

In our previous article, we explored the sufficient condition for the composition of covering maps to be a covering map, given the existence of an universal covering. In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights.

Q: What is the significance of the universal covering in this context?

A: The universal covering plays a crucial role in this context as it provides a "universal" space that covers all other covering spaces of the base space $B$. This allows us to establish a connection between the universal covering and the composition of covering maps.

Q: How does the universal covering ensure that the composition of covering maps is a covering map?

A: The universal covering ensures that the composition of covering maps is a covering map by providing a way to lift the maps to the universal covering space. This allows us to use the properties of the universal covering to establish the even covering property of the composition of maps.

Q: What is the even covering property, and why is it important?

A: The even covering property states that for any point $x$ in the space, there exists a neighborhood $U$ of $x$ such that the preimage of $U$ under the map is evenly covered by the map. This property is important because it ensures that the map is a covering map.

Q: How does the universal covering help to establish the even covering property of the composition of maps?

A: The universal covering helps to establish the even covering property of the composition of maps by providing a way to lift the maps to the universal covering space. This allows us to use the properties of the universal covering to establish the even covering property of the composition of maps.

Q: What are some implications of this result?

A: This result has several implications, including the fact that the composition of covering maps is a covering map if and only if the base space has a universal covering. This provides a new way to understand the properties of covering maps and their compositions.

Q: Can you provide some examples to illustrate this result?

A: Yes, here are a few examples:

• Let $B$ be the circle $S^1$ and let $p: X \rightarrow B$ be a covering map. Then the composition $p \circ p: X \rightarrow X$ is a covering map if and only if $X$ has a universal covering.
• Let $B$ be the torus $T^2$ and let $p: X \rightarrow B$ be a covering map. Then the composition $p \circ p: X \rightarrow X$ is a covering map if and only if $X$ has a universal covering.

Q: How does this result relate to other areas of mathematics?

A: This result has connections to other areas of mathematics, including algebraic topology and differential geometry. The concept of universal covering and covering maps is also important in the study of manifolds and their properties.

Q: What are some open questions and future directions for?

A: Some open questions and future directions for research include:

• Can we generalize this result to higher-dimensional spaces?
• What are the implications of this result for the study of manifolds and their properties?
• Can we develop new techniques for establishing the even covering property of the composition of maps?

Conclusion

In this article, we have provided a Q&A section to further clarify the concepts and provide additional insights into the sufficient condition for the composition of covering maps to be a covering map, given the existence of an universal covering. We hope that this article has been helpful in understanding this result and its implications.