Problem 9 - Algebraic Topology

Question

Compute the homology groups of the following 2-complexes: (a) The quotient of \(S^{2}\) obtained by identifying north and south poles to a point. (b) \(S^{1} \times\left(S^{1} \vee S^{1}\right)\) (c) The space obtained from \(D^{2}\) by first deleting the interiors of two disjoint subdisks in the interior of \(D^{2}\) and then identifying all three resulting boundary circles together via homeomorphisms preserving clockwise oricntations of these circles. (d) The quotient space of \(S^{1} \times S^{1}\) obtained by identifying points in the circle \(S^{1} \times\left\\{x_{0}\right\\}\) that differ by \(2 \pi / m\) rotation and identifying points in the circle \(\left\\{x_{0}\right\\} \times S^{1}\) that differ by \(2 \pi / n\) rotation.

Step-by-Step Solution

Answer

(a) \(H_0 = \mathbb{Z}\), \(H_1 = \mathbb{Z}/2\mathbb{Z}\), \(H_2 = 0\). (b) \(H_0 = \mathbb{Z}\), \(H_1 = \mathbb{Z}^3\), \(H_2 = \mathbb{Z}\). (c) \(H_0 = \mathbb{Z}\), \(H_1 = \mathbb{Z}^3\), \(H_2 = 0\). (d) \(H_0 = \mathbb{Z}\), \(H_1 = \mathbb{Z}^2\), \(H_2 = \mathbb{Z}\).

Step 1: Understanding the given space (a)

The quotient of \(S^2\) by identifying the north and south poles to a point results in a space known as \(\mathbb{RP}^2\), the real projective plane.

Step 2: Compute homology of \(\mathbb{RP}^2\) (a)

The homology groups of \(\mathbb{RP}^2\) are well-known: - \(H_0(\mathbb{RP}^2) \cong \mathbb{Z}\), - \(H_1(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z}\), - \(H_2(\mathbb{RP}^2) = 0\).

Step 3: Understanding the given space (b)

The space \(S^1 \times (S^1 \vee S^1)\) is a 2-dimensional torus with two additional disjoint circles joined at a point, resembling a 'figure-eight' on top of the torus.

Step 4: Compute homology of \(S^1 \times (S^1 \vee S^1)\) (b)

For this space, the homology groups can be computed as: - \(H_0(S^1 \times (S^1 \vee S^1)) \cong \mathbb{Z}\), - \(H_1(S^1 \times (S^1 \vee S^1)) \cong \mathbb{Z}^3\), - \(H_2(S^1 \times (S^1 \vee S^1)) \cong \mathbb{Z}\).

Step 5: Understanding the given space (c)

Start with a disk \(D^2\), remove two interior disks, and then identify the three resulting boundary components together, which topologically results in a 2-sphere with two points identified, giving rise to a space equivalent to the wedge of three circles, \(S^1 \vee S^1 \vee S^1\).

Step 6: Compute homology of \(S^1 \vee S^1 \vee S^1\) (c)

The homology of \(S^1 \vee S^1 \vee S^1\) is:- \(H_0(S^1 \vee S^1 \vee S^1) \cong \mathbb{Z}\), - \(H_1(S^1 \vee S^1 \vee S^1) \cong \mathbb{Z}^3\), - \(H_2(S^1 \vee S^1 \vee S^1) = 0\).

Step 7: Understanding the given space (d)

In this problem, the surface \(T^2 = S^1 \times S^1\) is being identified along its curves similar to turning it into a torus with modified slopes, leading to a torus with a twist at each marked point.

Step 8: Compute homology with rotations (d)

The operation about axes properly retains the torus’s topology, yielding: - \(H_0(T^2) \cong \mathbb{Z}\), - \(H_1(T^2) \cong \mathbb{Z}^2\), - \(H_2(T^2) \cong \mathbb{Z}\).

Key Concepts

Homology Groups

Homology groups are mathematical structures used in algebraic topology to analyze and classify the different types of structures and holes within a topological space. These groups are denoted as \(H_n(X)\), where \(X\) is a topological space and \(n\) represents the dimension. Homology groups give insight into the number and type of "holes" in each dimension.

• \(H_0(X)\) represents the path-connected components in the space. It gives the number of connected pieces.
• \(H_1(X)\) describes 1-dimensional "holes", similar to loops or tunnels.
• \(H_2(X)\) corresponds to 2-dimensional "holes", often visualized like cavities within the space.

For instance, if a surface has three loops, its first homology group, \(H_1\), would represent these loops. Each new homology group corresponds to a different level of complexity, all of which are integral in understanding the topology of any given space.

2-Complexes

A 2-complex is a type of topological space that is built using 2-dimensional polygons, like disks. These spaces can often be used to represent surfaces and are foundational in algebraic topology. They are formulated by connecting edges of these polygons, and they can be analyzed using their homology.
For example, when polygons in a 2-complex are glued together, new shapes and surfaces are formed.

• These complexes can help visualize and understand complicated surfaces like the torus or the real projective plane.
• Through their study, one can determine properties like connectedness and the number of holes.

Complexes are surprisingly versatile, giving rise to various kinds of spaces by interpreting how different polygons are put together.

Real Projective Plane

The Real Projective Plane, denoted as \(\mathbb{RP}^2\), is an important geometric surface in algebraic topology. It can initially seem abstract and complex, yet with further exploration, its properties become fascinating.

• The plane can be visualized as a disk with opposite points on the boundary identified.
• Another way to imagine \(\mathbb{RP}^2\) is by taking a sphere and identifying opposing points as single points, like joining the north and south poles.
• Notably, this space is non-orientable, meaning there's no consistent way to differentiate between the 'inside' and 'outside'.

When analyzing its homology groups, \(\mathbb{RP}^2\) has some interesting properties. For instance, \(H_1(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z}\), reflecting its non-orientability, and confirming the existence of a single type of twist or loop within the structure.

Torus

The torus is a doughnut-shaped surface that is, in many ways, a classic example in topology due to its simple yet rich structure. It can be represented as \(S^1 \times S^1\), where \(S^1\) is the circle.

• The torus has both a major and a minor circumference: representing rotations around its center and along its tube.
• Unlike the real projective plane, the torus is orientable; one can consistently define what is 'inside' and 'outside'.
• Topologically, the torus is significant because it features multiple holes and loops.

Analyzing its homology groups reveal that, for the torus \(H_1(T^2) \cong \mathbb{Z}^2\), displaying the two fundamental loops through its structure. Additionally, its second homology group, \(H_2(T^2) \cong \mathbb{Z}\), represents a cavity enclosed by the object, underscoring the torus's universal relevance in studies of surface topology.