Torus

Definition

A torus is a doughnut-shaped surface that can be mathematically represented as the Cartesian product of two circles, denoted as $S^1 \times S^1$. This shape is fundamental in algebraic topology, serving as a classic example of a non-trivial topological space that can help illustrate various concepts such as homotopy, homology, and the structure of CW complexes.

5 Must Know Facts For Your Next Test

1. The torus can be visualized as a rectangle with opposite edges identified; this helps in understanding its topological properties.
2. In terms of homology, the torus has interesting properties: it has a non-trivial first homology group, which is $ ext{H}_1(S^1 \times S^1) \cong \mathbb{Z} \oplus \mathbb{Z}$.
3. The fundamental group of the torus is $ ext{π}_1(T^2) \cong \mathbb{Z} \oplus \mathbb{Z}$, indicating that it has loops that cannot be continuously shrunk to a point without leaving the torus.
4. The Mayer-Vietoris sequence can be used to compute the homology groups of the torus by breaking it down into simpler pieces.
5. The torus is an important example in algebraic topology as it illustrates concepts like homotopy equivalence between spaces.

Review Questions

• How can you utilize the Mayer-Vietoris sequence to understand the homology of a torus?
  • The Mayer-Vietoris sequence allows us to compute the homology groups of the torus by decomposing it into two overlapping parts. By analyzing these subspaces and their intersection, we can derive information about the entire space's homology. This method reveals how complex shapes like the torus can be understood through simpler ones and highlights their topological properties.
• Describe how the fundamental group of a torus differs from that of a sphere and what implications this has for their respective topological properties.
  • The fundamental group of a torus is $ ext{π}_1(T^2) \cong \mathbb{Z} \oplus \mathbb{Z}$, which indicates that it has multiple non-trivial loops that cannot be shrunk to a point. In contrast, the fundamental group of a sphere is trivial, $ ext{π}_1(S^2) = 0$, meaning all loops can be contracted. This difference in fundamental groups highlights the torus's more complex topology and shows how it has 'holes' compared to the simply connected sphere.
• Evaluate the significance of CW complexes in relation to constructing and analyzing the torus, particularly in terms of cellular homology.
  • CW complexes provide an effective way to construct topological spaces like the torus using cells glued together. By viewing the torus as a CW complex with two 1-cells and one 2-cell, we can apply cellular homology to compute its homology groups easily. This method not only simplifies calculations but also deepens our understanding of how spaces can be built from basic components while preserving their essential topological features.

Related terms

Homotopy:

A continuous deformation of one function into another within a topological space, allowing for the classification of spaces based on their shape.

Fundamental Group: An algebraic structure that encodes the looped paths in a space, reflecting its shape and allowing the classification of topological spaces based on their connectivity.

CW Complex: A type of topological space constructed from cells (disks) glued together in a specific way, used to study homology and other topological properties.