Dimension

Definition

Dimension refers to the minimum number of coordinates needed to specify a point within a mathematical space. It serves as a fundamental concept in topology and geometry, allowing us to classify spaces based on their complexity and structure. The concept of dimension connects various important features, such as the behavior of submanifolds, the intricacies of embeddings, and the properties of different types of manifolds like spheres and tori.

5 Must Know Facts For Your Next Test

1. The dimension of a manifold is defined as the number of local coordinates needed to describe points in it; for example, a line has dimension 1, a plane has dimension 2, and so forth.
2. In the context of submersions, regular values relate to the mapping's dimensions, where the dimension of the domain minus the dimension of the image gives insights into critical points.
3. Spheres, tori, and projective spaces are examples of different manifolds with distinct dimensions; for instance, a sphere in three-dimensional space has dimension 2.
4. The Whitney Embedding Theorem highlights that any n-dimensional manifold can be embedded into R^(2n), indicating how dimensions interact in embeddings.
5. Understanding dimension helps in classifying spaces and analyzing how different topological properties, such as compactness or connectivity, behave across varying dimensions.

Review Questions

• How does the concept of dimension affect the classification of manifolds?
  • Dimension plays a crucial role in classifying manifolds because it determines how many local coordinates are required to describe them. For example, a 1-dimensional manifold behaves differently than a 2-dimensional manifold. This distinction is significant when analyzing properties such as compactness or connectedness since these can vary drastically based on dimensionality.
• Discuss how submersions and regular values relate to the concept of dimension in differential topology.
  • In differential topology, submersions are smooth maps where the rank equals the dimension of the domain at every point. Regular values are those that have pre-images consisting only of regular points. The relationship between these concepts highlights how dimensions dictate the behavior of mappings between manifolds. Specifically, understanding the dimensions involved helps identify critical points where the behavior of these mappings changes.
• Evaluate the significance of the Whitney Embedding Theorem concerning dimensions in topology.
  • The Whitney Embedding Theorem's significance lies in its assertion that any n-dimensional smooth manifold can be embedded into R^(2n). This has profound implications for understanding how manifolds interact with Euclidean spaces of various dimensions. By showing that manifolds can be represented within higher-dimensional spaces, it provides tools for visualizing and analyzing their properties while reinforcing the importance of dimension in topology.

Related terms

Manifold: A manifold is a topological space that locally resembles Euclidean space near each point, allowing for the generalization of concepts like curves and surfaces to higher dimensions.

Submanifold:

A submanifold is a subset of a manifold that is itself a manifold, inheriting its structure and dimension from the larger manifold.

Whitney Embedding Theorem:

The Whitney Embedding Theorem states that any smooth manifold can be embedded into a Euclidean space of sufficiently high dimension, providing insights into the relationship between manifolds and their dimensions.