Problem 29 - Algebraic Topology

Question

Show that \(S^{1} \times S^{1}\) and \(S^{1} \vee S^{1} \vee S^{2}\) have isomorphic homology groups in all dimensions, but their universal covering spaces do not.

Step-by-Step Solution

Answer

The homology groups of \(S^1 \times S^1\) and \(S^1 \vee S^1 \vee S^2\) are isomorphic, but their universal covers are not.

Step 1: Define Homology Groups for \(S^1 \times S^1\)

The space \( S^1 \times S^1 \) is a torus. For a torus, the homology groups are:\[H_0(S^1 \times S^1) \cong \mathbb{Z},H_1(S^1 \times S^1) \cong \mathbb{Z} \oplus \mathbb{Z},H_2(S^1 \times S^1) \cong \mathbb{Z},H_n(S^1 \times S^1) \cong 0 \text{ for } n \geq 3.\]

Step 2: Define Homology Groups for \(S^1 \vee S^1 \vee S^2\)

The space \( S^1 \vee S^1 \vee S^2 \) (a wedge of two circles and a sphere) has the following homology groups:\[H_0(S^1 \vee S^1 \vee S^2) \cong \mathbb{Z},H_1(S^1 \vee S^1 \vee S^2) \cong \mathbb{Z} \oplus \mathbb{Z},H_2(S^1 \vee S^1 \vee S^2) \cong \mathbb{Z},H_n(S^1 \vee S^1 \vee S^2) \cong 0 \text{ for } n \geq 3.\]

Step 3: Compare Homology Groups

Comparing the previously derived homology groups of both spaces, we observe:- \(H_0\) for both spaces is \(\mathbb{Z}\).- \(H_1\) for both spaces is \(\mathbb{Z} \oplus \mathbb{Z}\).- \(H_2\) for both spaces is \(\mathbb{Z}\).- \(H_n\) is zero for both spaces for \( n \geq 3 \).Thus, \(S^1 \times S^1\) and \(S^1 \vee S^1 \vee S^2\) have isomorphic homology groups in all dimensions.

Step 4: Consider Universal Covers

The universal cover of \( S^1 \times S^1 \) is \( \mathbb{R}^2 \), as it is a 2-dimensional manifold with no holes, making all its homology groups zero, except \( H_0(\mathbb{R}^2) \cong \mathbb{Z} \).For \( S^1 \vee S^1 \vee S^2 \), the universal cover consists of infinite copies of \( S^2 \), because its fundamental group is a free group on two generators; it doesn't contract to a point. Each \( S^2 \) contributes \( H_2 \cong \mathbb{Z} \).

Step 5: Compare Universal Covers

Comparing universal covers:- Universal cover of \(S^1 \times S^1\): \(H_0(\mathbb{R}^2) \cong \mathbb{Z}\), \(H_n(\mathbb{R}^2) = 0\) for all \(n \geq 1\).- Universal cover of \(S^1 \vee S^1 \vee S^2\): \(H_2\) is non-trivial (\(\mathbb{Z}\)), unlike any homology group of the universal cover of \(S^1 \times S^1\).Therefore, their universal covers are not homologically equivalent.

Key Concepts

Homology Groups

Homology groups are algebraic constructs that allow us to study topological spaces using the language of algebra. They provide a set of groups for each dimension, capturing essential information about a space's structure, such as holes and voids. For the torus, denoted as \(S^1 \times S^1\), the homology groups are:

• \(H_0(S^1 \times S^1) \cong \mathbb{Z}\)
• \(H_1(S^1 \times S^1) \cong \mathbb{Z} \oplus \mathbb{Z}\)
• \(H_2(S^1 \times S^1) \cong \mathbb{Z}\)
• \(H_n(S^1 \times S^1) \cong 0\) for \(n \geq 3\)

The wedge sum of two circles and a sphere, \(S^1 \vee S^1 \vee S^2\), has strikingly similar groups:

• \(H_0(S^1 \vee S^1 \vee S^2) \cong \mathbb{Z}\)
• \(H_1(S^1 \vee S^1 \vee S^2) \cong \mathbb{Z} \oplus \mathbb{Z}\)
• \(H_2(S^1 \vee S^1 \vee S^2) \cong \mathbb{Z}\)
• \(H_n(S^1 \vee S^1 \vee S^2) \cong 0\) for \(n \geq 3\)

This means both spaces have the same homology groups, suggesting they are similar in a homological sense.

Universal Covering Spaces

The universal covering space of a topological space is, essentially, the most simplified version of the space that covers it without any "twists" or "holes." Every loop on the universal cover can be shrunk to a point but preserves the same number of copies of the original space. For example, the universal cover of a torus \(S^1 \times S^1\) is \(\mathbb{R}^2\).

• Homology of \(\mathbb{R}^2\): \(H_0(\mathbb{R}^2) \cong \mathbb{Z}\), \(H_n(\mathbb{R}^2) = 0\) for all \(n \geq 1\)

In contrast, the universal cover of \(S^1 \vee S^1 \vee S^2\) is composed of infinitely many spheres \(S^2\), due to its fundamental group's nature. Loop patterns don't "wrap around," resulting in repetitive copies.

• Homology of infinite spheres: \(H_2 \cong \mathbb{Z}\), showing non-trivial nature in dimension 2

This divergence in the universal covering spaces leads to different homological properties.

Torus

A torus, denoted \(S^1 \times S^1\), is essentially the shape of a doughnut. It's formed by revolving a circle around an axis, which creates a cylindrical structure with a hole. This shape gives it rich topological properties.The fundamental feature of the torus is its two loops: one along the circle of the torus and another around the hole. These loops correspond to the generators of its homology:

• First circle (meridian) representing \(H_1\)
• Second circle (longitude) representing another part of \(H_1\)

Together, these form \(\mathbb{Z} \oplus \mathbb{Z}\), reflecting the idea of moving in two independent directions.Due to its structure, the torus's universal cover \(\mathbb{R}^2\) is flat without holes, further emphasizing how the torus itself can "tile" space like a grid.

Wedge Sum

The wedge sum, often denoted by the symbol \(\vee\), is a method to combine topological spaces at a single shared point. For example, \(S^1 \vee S^1 \vee S^2\) combines two circles and a sphere.In forming a wedge sum, each component's independent loops are preserved, giving rise to new homology characteristics:

• Each circle \(S^1\) contributes to \(H_1\)
• The sphere \(S^2\) affects \(H_2\)

Such constructions emphasize the additive nature of homology: components "add up" contributing their respective features to the greater whole. This becomes particularly significant in recognizing universal covers, where distinct elements create a composite path structure beyond initial appearances.

Fundamental Group

The fundamental group, denoted \(\pi_1\), is the topological space's "loop space." It provides insight into the geometry of a space by examining the loops beginning and ending at a single point. For \(S^1 \times S^1\), this group is \(\mathbb{Z} \times \mathbb{Z}\) because it can repeatedly "wrap" around two independent circles.For \(S^1 \vee S^1 \vee S^2\), the fundamental group is more complex, represented as a free group on two generators. This free group property implies intertwining loops that cover an infinite spectrum of arrangements.Understanding the fundamental group allows us to predict universal covers:

• The torus's fundamental group indicates a simple, repeating structure (\(\mathbb{R}^2\))
• The wedge sum's free group structure suggests repeat copies (infinite \(S^2\))

Fundamental groups thus provide a foundation for predicting and understanding more advanced properties like those in homology and universal covers.