Cosmic Topology: A Geometric Approach

The introductory section sets the stage by contrasting Euclidean geometry with non-Euclidean geometries, specifically highlighting hyperbolic and elliptic geometries as the primary focus. It emphasizes that the text aims to narrate a mathematical story rather than function as a mere reference. The pedagogical approach includes class discussions and interactive activities, like the Coneland and Saddleland examples (1.3.5 and 1.3.7), designed to enhance understanding. The importance of active learning, including working through examples and exercises, is repeatedly stressed. The reader is encouraged to engage actively, using tools like compass and ruler constructions, or software such as The Geometer's Sketchpad or Geogebra. This hands-on approach underscores the importance of visualization and intuitive grasp alongside theoretical understanding. The use of constructions within proofs and definitions is also highlighted. The section clearly establishes the importance of actively engaging with the material.

A significant emphasis is placed on the use of compass and ruler constructions, emphasizing inversions as foundational building blocks of transformations. These constructions are not merely illustrative; they are integral to proofs, such as the Fundamental Theorem of Möbius Transformations, and serve as guides for defining key concepts. The text encourages readers to perform constructions themselves, using physical tools or software, promoting a deeper understanding. The availability of additional Geometer's Sketchpad templates and activities on the text's website further supports this active learning approach. The text contrasts this active, hands-on approach with the potential for online readers to passively jump between sections, advocating for a linear, slower reading experience to fully grasp the unfolding mathematical narrative. The inherent connection between visual construction and theoretical understanding of Möbius transformations is central.

The introduction employs an analogy of a two-dimensional being inhabiting a two-dimensional universe to illustrate fundamental geometric principles. This two-dimensional world, initially presented as an infinite plane similar to the xy-plane in calculus, serves as a familiar reference point for Euclidean geometry. The familiar property of triangle angles summing to 180° in Euclidean geometry is introduced. The text then uses the example of builders using the Pythagorean theorem to highlight the practical applications and the homogeneity of Euclidean geometry. This homogeneous nature, where local geometry is consistent across all points, is contrasted with non-homogeneous examples later in the text. The section cleverly builds a foundation of familiar Euclidean concepts to contrast against the less intuitive aspects of non-Euclidean geometries that are introduced later. The introduction of a finite two-dimensional world with no boundary (Figure 1.1.2), represented as a rectangle with identified edges, paves the way for the discussion of surfaces and the construction of a torus.

The historical context surrounding the development of non-Euclidean geometry is introduced, highlighting the work of János Bolyai and Nikolai Lobachevsky (1792-1856), who independently demonstrated that Euclid's fifth postulate was not a necessary consequence of the first four. This pivotal discovery fundamentally altered the landscape of geometry. The section notes the impact of non-Euclidean geometry extending beyond the mathematics community, mentioning its inclusion in Fyodor Dostoevsky's The Brothers Karamazov (published in 1880). This literary reference emphasizes the broader intellectual and philosophical implications of the shift from a solely Euclidean worldview. Ivan Karamazov's use of non-Euclidean geometry to illustrate the limits of human understanding of complex concepts like the existence of God provides a striking example of the impact of this geometric revolution. The section subtly connects the mathematical breakthroughs to broader intellectual and philosophical discussions.

The introduction proceeds to examine the concept of homogeneity in geometric spaces. It contrasts the homogeneous nature of Euclidean geometry on a plane (where local geometry remains consistent everywhere) with the non-homogeneous nature of a donut-shaped surface. This leads to the introduction of the 'flat torus' (Example 1.3.4), a homogeneous surface where local geometry is Euclidean everywhere despite its global topology. The analogy of a ship's pilot on a flat torus is used to illustrate the finite but boundary-less nature of this space and the surprising consequence of being able to see the back of one's own ship (or even head!). This example effectively uses a visual metaphor to transition from simple Euclidean ideas to more complex concepts of non-Euclidean geometries and surfaces. The contrast between local and global properties becomes a central theme, influencing subsequent sections focusing on different geometric models.

This section delves into how a two-dimensional creature inhabiting a surface could determine the underlying geometry of its world. The key approach is to examine the properties of simple geometric shapes: triangles and circles. The sum of angles in a triangle serves as a crucial indicator. In Euclidean geometry, this sum is always 180°. However, in non-Euclidean geometries, this sum deviates—being less than 180° in hyperbolic geometry and greater than 180° in elliptic geometry. The relationship between the circumference and radius of a circle also provides valuable clues. While in Euclidean geometry, the circumference is approximately 2πr, this relationship changes noticeably for larger circles in non-Euclidean spaces, indicating the curvature of the surface. The section uses the examples of Coneland (positive curvature) and Saddleland (negative curvature) to illustrate how the angle sum of triangles varies across different geometries. The concept of a geodesic, representing the shortest path between two points on a surface, is also introduced, refining the definition of a circle on a curved surface.

The concept of a flat torus is introduced as a specific example of a surface with Euclidean geometry. The flat torus is constructed by taking a rectangle and identifying opposite edges, effectively creating a two-dimensional surface that is both finite and without boundaries. The text uses the analogy of a ship's pilot on this surface; from any point, the local geometry appears Euclidean (triangle angles add to 180°), but globally, the space is finite. This allows the pilot to potentially see the back of their ship or even their own head with a powerful enough telescope, demonstrating the unusual global properties of this space. The flat torus serves as a powerful example of how a space can be locally Euclidean yet globally distinct from an infinite Euclidean plane. This example beautifully illustrates the difference between local and global properties of surfaces, a crucial concept in the study of geometry and topology.

The section emphasizes the properties of homogeneity and isotropy in characterizing the geometry of surfaces. A surface is homogeneous if its local geometry is the same at every point; it is isotropic if the geometry is the same in every direction at each point. The text highlights that a two-dimensional bug can use small triangles and their angle sums to distinguish between different points on a surface, thus revealing whether the surface is homogeneous or not. The example of a bug on a donut-shaped surface illustrates this concept: triangles in different regions will have varying angle sums, thus revealing the surface's non-homogeneity. The section uses the example of a bug performing experiments to determine if its world is homogeneous or not and even ponders the possibility of a Nobel prize for this groundbreaking discovery! This reinforces the practical implications of understanding these properties for distinguishing between different types of geometries on surfaces.

This section introduces hyperbolic geometry, focusing on the Poincaré disk model developed by Henri Poincaré (1854-1912). The text highlights Poincaré's diverse interests, including the intersection of mathematics, physics, and psychology, and notes that his work on non-Euclidean geometry stemmed from his studies of differential equations and number theory. Poincaré's perspective, influenced by Klein's view of geometries as generated by sets and groups of transformations, is mentioned. The section establishes the Poincaré disk model as the primary focus for exploring hyperbolic geometry, briefly mentioning the existence of another model—the upper half-plane model—which will be addressed later. The introduction sets the stage by emphasizing Poincaré's significant contribution to the field and the chosen model's importance in the text's exploration of hyperbolic geometry.

The discussion then delves into the properties of Möbius transformations within the context of hyperbolic geometry. The concept of clines of inversion is introduced, and it is explained that if two clines of inversion do not intersect, they define two fixed points of the resulting Möbius transformation, which lie on the unit circle. The transformation is then classified as a translation of the hyperbolic plane. The text also touches upon the invariance of distances under transformations, essential for defining a distance function in hyperbolic geometry. This section builds on previously introduced concepts of Möbius transformations, emphasizing their role in the study of hyperbolic geometry. The introduction of fixed points and translations sets the framework for a deeper understanding of transformations and symmetries within this non-Euclidean space. The discussion subtly foreshadows the development of a distance function, a critical aspect of any geometric model.

A key aspect of any geometric system is the ability to measure distances, angles, and areas. This section highlights how these measurements are conducted in the Poincaré disk model of hyperbolic geometry. The crucial difference regarding triangles is emphasized: in hyperbolic geometry, the sum of angles in a triangle is always less than 180°, a stark contrast to Euclidean geometry. The degree of this difference depends on the size of the triangle, with larger triangles exhibiting more significant deviations from the 180° sum. This is illustrated using examples (∆zuw and ∆pqr), demonstrating how the angle deficiency becomes more pronounced as triangles grow larger. The text also mentions the development of a distance function that remains invariant under hyperbolic transformations, preparing the groundwork for subsequent discussions of hyperbolic metrics. The emphasis on the angle deficiency in triangles and the invariance of the distance function under transformations highlights the fundamental differences between hyperbolic and Euclidean geometries.

The section introduces a method for constructing a four-sided figure that resembles a rectangle in hyperbolic geometry. It is crucial to note that this construction uses hyperbolic lines for only two opposite sides; the other sides are cline arcs. This construction demonstrates the limitations of applying Euclidean notions directly to hyperbolic spaces. The construction involves choosing points on the positive real axis and utilizing hyperbolic lines perpendicular to this axis. The resulting figure has right angles and opposite sides of equal length, though it deviates from the Euclidean notion of a rectangle as all four sides aren't hyperbolic segments. An exercise (referencing Example 5.4.17) is proposed, involving a bug tracing a path resembling a square in Euclidean space but whose characteristics will be measured using hyperbolic triangle trigonometry. This exercise highlights that even though the construction is inspired by Euclidean shapes, the resulting figure's properties are analyzed within the context of hyperbolic trigonometry, further distinguishing hyperbolic from Euclidean geometry.

This section introduces elliptic geometry, using the two-dimensional surface of a sphere as its primary model. The text emphasizes that this choice of the sphere as a model is not arbitrary, as it deeply impacts geometric properties. The chapter starts by reviewing stereographic projection—a mapping technique used to transfer information from the sphere onto the extended complex plane. This mapping facilitates translating geometric features of the spherical model into a more readily analyzed planar representation. The use of stereographic projection highlights the importance of using appropriate mappings to analyze geometric properties within the model. The section lays the groundwork for understanding elliptic geometry by establishing the sphere as a fundamental model and introducing the tool of stereographic projection.

The section defines distances and lines within the elliptic geometry model. It highlights that between any two points on the sphere (or projective plane), there exists a unique elliptic line connecting them; however, this line can be viewed as consisting of two segments, both connecting the points. This is because traveling along this elliptic line could be done in two directions. The elliptic distance between the points is thus defined as the length of the shorter of the two segments. The concept of elliptic distance is further illustrated using the example of the distance between points 0 and 1, equating to π/2, corresponding to one-quarter of a great circle on the unit sphere. The distance formula is provided and verified against measurements on the unit 2-sphere, with an exercise (Exercise 6.3.12) suggested to further explore this relationship. The section clearly introduces the concept of elliptic distance and how it differs from Euclidean distance, highlighting the unique characteristics of elliptic lines on the sphere.

The section discusses how the postulates of geometry, particularly Euclid's postulates, are satisfied within the elliptic geometry model using the sphere (or the projective plane) as its representation. It is explicitly stated that even though elliptic space is finite (due to the sphere's bounded nature), line segments can be extended indefinitely because the space has no boundary. This subtle detail is significant, emphasizing the difference between a finite space and one with boundaries. The discussion then relates to the projective plane (P²), a surface obtained by identifying antipodal points on the sphere. The text highlights that the first postulate of Euclidean geometry holds in the projective plane (Theorem 6.2.11) and mentions that the second postulate also holds despite the finite nature of the space. The section highlights that while the space is finite, it still lacks boundaries, a fundamental difference that impacts how geometric principles operate within this space.

This section establishes a crucial link between arc length and the metric on surfaces, demonstrating that arc length ensures the shortest path between two points is along the line connecting them. This principle holds true across different geometries (hyperbolic, Euclidean, elliptic) and for arbitrary values of a constant k, providing a general metric applicable to various geometric spaces. The arc length formula defines a distance function, dₖ(p, q), which represents the length of the shortest path between points p and q. A circle in this context is defined as the set of points equidistant from a central point, using this distance function. The concept of homogeneity and isotropy is also introduced; a space is homogeneous if its local geometry is the same at every point and isotropic if it's the same in every direction at each point. The section lays a solid foundation for understanding metrics and how distances are defined and measured across diverse geometrical contexts.

The section contrasts topological and geometric features of surfaces. Topological features remain unchanged under continuous deformations (stretching, bending), while geometric features (volume, curvature, surface area) change. The text uses the example of a ball: no matter how it is deformed, a loop drawn on its surface separates it into two pieces (topological property), but its volume and curvature change (geometric properties). The distinction between continuous deformation (topology) and changes in metric properties (geometry) is highlighted. This introduces the idea that certain properties are invariant under topological transformations while others are not, setting the scene for classifying surfaces based on properties unaffected by continuous deformations.

The section delves into the classification of surfaces using connected sums and the Euler characteristic. Connected sums are operations where two surfaces are joined by removing a disk from each and gluing their boundaries. The text uses the example of the torus (T²) and the sphere (S²) to demonstrate this concept, with the sphere acting as a neutral element (S² # X = X). The concept of a cross-cap is introduced; a cross-cap surface Cg is topologically equivalent to a sphere with g open 2-balls replaced with cross-caps. The text uses C₁ to represent the projective plane. A method is introduced to represent surfaces using polygons and their edge identifications. This method uses boundary labels to encode the edge identifications of a polygon, demonstrating how the Euler characteristic, related to the surface's curvature, dictates the type of homogeneous geometry the surface can admit. The text connects the topological concept of the Euler characteristic to the geometric property of curvature.

This section ties together the Euler characteristic, curvature, and the potential shapes of a universe. The text states that surfaces admitting one of the three geometries (Euclidean, hyperbolic, elliptic) have constant curvature. The sign of this curvature matches the sign of the surface's Euler characteristic. While the type of geometry is determined by the Euler characteristic (and hence the shape), the curvature's magnitude can change by scaling (e.g., changing the radius of a sphere). The discussion also extends to a two-dimensional cosmologist who could determine the shape of their universe by measuring curvature and area, thereby deducing the Euler characteristic. Knowing the Euler characteristic narrows down the possible shapes of the universe: a characteristic of 2 or an odd integer uniquely defines the shape, while an even characteristic less than 2 leaves two possibilities (one orientable, one non-orientable). The section closes by tying together the mathematical concepts discussed previously to explore the potential shapes of the universe, illustrating a significant application of these geometrical concepts.

This section introduces the concept of three-dimensional geometry and 3-manifolds, extending the principles discussed earlier for two-dimensional surfaces. It's highlighted that, similar to two-dimensional surfaces, three-dimensional spaces can admit Euclidean, hyperbolic, or elliptic geometries. The text explains that constructing 3-manifolds can involve identifying faces of a 3-complex (such as Platonic solids) in pairs. The choice of how faces are identified determines the resulting geometry of the 3-manifold. The section serves as a bridge, extending the established principles of two-dimensional geometries to the three-dimensional realm, setting the stage for a discussion of specific examples of 3-manifolds and their associated geometries.

The section delves into the classification of elliptic 3-manifolds, stating that there are infinitely many different types, all of which are orientable. The simplest elliptic 3-manifold, the 3-sphere, is discussed in relation to Albert Einstein's early model of the universe. Einstein favored this model due to its static, finite, simply connected nature, without boundaries. However, the text emphasizes that general relativity only dictates the local nature of space, leaving the global shape open. The Dutch astronomer Willem de Sitter (1872–1934) expanded on this, proposing a different global shape: a three-dimensional elliptic space obtained by identifying antipodal points on the 3-sphere, a concept similar to the earlier discussions about projective planes. The discussion illustrates how the mathematical properties of 3-manifolds are related to cosmological models of the universe, specifically mentioning Einstein's early assumptions.

This section provides specific examples of 3-manifolds and the geometries they inherit. The 3-torus, a three-dimensional analog of the previously discussed torus, is described as inheriting Euclidean geometry because its corner angles perfectly fit together under face identification. The Poincaré dodecahedral space, another example, is presented as a potential model for the shape of our universe. This model is constructed from a dodecahedron with its opposite faces identified with a one-tenth clockwise twist. The text also describes the construction of lens spaces from the unit solid ball by identifying points on its boundary 2-sphere, and mentions the Hantschze-Wendt manifold, which is constructed from two cubes sharing a face and inherits Euclidean geometry. These examples provide concrete illustrations of how different 3-manifolds can be constructed and the types of geometries they admit, showcasing the diversity of three-dimensional spaces and their possible geometric properties.

This section bridges the abstract concepts of geometry and topology with the real-world implications for cosmology. It discusses how the global geometry of the universe might be non-Euclidean, implying that our three-dimensional space could be curving in some unseen fourth dimension. Under reasonable assumptions, the text states that hyperbolic, elliptic, and Euclidean geometries are the only three possibilities for the global geometry of our universe. The challenge of determining which geometry applies to our universe using cosmological data is discussed. The connection between the universe's geometry and its finite or infinite nature is also explored; for example, an elliptic universe would necessarily be finite in volume. This section firmly grounds the theoretical framework in the realm of cosmology, posing the crucial question of the actual geometry of our universe.

The section explores how geometric principles can be applied to cosmological observations. It uses the example of a torus-shaped universe to illustrate the challenges and possibilities. The concept of tiling the plane with identical copies of the torus is introduced to visualize how objects in such a universe would appear. By considering the observable universe's diameter (2r_obs), the text examines the condition under which multiple images of the same object could be observed. This relates to the size of the observable universe relative to the universe's overall dimensions. In the context of the torus, this method reveals that multiple images of an object would be visible. The section acknowledges that the evolution of galaxies over time would complicate recognition of such repeated images. The text acknowledges the challenges of identifying repeated images in a torus-shaped universe due to galaxy evolution and ultimately concludes that while the idea of seeing repeated images of our galaxy is intriguing, current size estimates make this unlikely.