Manifolds and the Shape of the Universe

Home , Euclidean space, Space (mathematics), Sphere, Torus

Stacy Hoehn

July 13, 2010

Stacy Hoehn Manifolds and the Shape of the Universe Stretching, shrinking, bending, and twisting are allowed.

Examples of Homeomorphic Objects:

What Does “Shape” Mean to a Topologist?

In topology, two objects have the same shape if one can be deformed into the other without cutting or gluing. Objects with the same shape are called homeomorphic.

Stacy Hoehn Manifolds and the Shape of the Universe Examples of Homeomorphic Objects:

What Does “Shape” Mean to a Topologist?

In topology, two objects have the same shape if one can be deformed into the other without cutting or gluing. Objects with the same shape are called homeomorphic.

Stretching, shrinking, bending, and twisting are allowed.

Stacy Hoehn Manifolds and the Shape of the Universe What Does “Shape” Mean to a Topologist?

In topology, two objects have the same shape if one can be deformed into the other without cutting or gluing. Objects with the same shape are called homeomorphic.

Stretching, shrinking, bending, and twisting are allowed.

Examples of Homeomorphic Objects:

Stacy Hoehn Manifolds and the Shape of the Universe A torus and a sphere are not homeomorphic.

Homeomorphic or Not?

A doughnut and a coﬀee cup are homeomorphic.

Stacy Hoehn Manifolds and the Shape of the Universe Homeomorphic or Not?

A doughnut and a coﬀee cup are homeomorphic.

A torus and a sphere are not homeomorphic.

Stacy Hoehn Manifolds and the Shape of the Universe 2 Locally, they both look two-dimensional Euclidean space R .

The torus and the sphere are both called 2-manifolds because they share this property.

2-Manifolds

Even though the torus and sphere are not homeomorphic, they do have something in common.

Stacy Hoehn Manifolds and the Shape of the Universe The torus and the sphere are both called 2-manifolds because they share this property.

2-Manifolds

Even though the torus and sphere are not homeomorphic, they do have something in common.

2 Locally, they both look two-dimensional Euclidean space R .

Stacy Hoehn Manifolds and the Shape of the Universe 2-Manifolds

Even though the torus and sphere are not homeomorphic, they do have something in common.

2 Locally, they both look two-dimensional Euclidean space R .

The torus and the sphere are both called 2-manifolds because they share this property.

Stacy Hoehn Manifolds and the Shape of the Universe Examples: 1-manifolds:

2-manifolds:

3 3-manifolds: R and the universe

n-Manifolds

Deﬁnition An n-manifold is a topological space that locally looks like n n-dimensional Euclidean space R .

Stacy Hoehn Manifolds and the Shape of the Universe n-Manifolds

Deﬁnition An n-manifold is a topological space that locally looks like n n-dimensional Euclidean space R .

Examples: 1-manifolds:

2-manifolds:

3 3-manifolds: R and the universe

Stacy Hoehn Manifolds and the Shape of the Universe The Surface of the Earth

Locally, the surface of the Earth looks like a 2-dimensional plane, so it is a 2-manifold. If we only saw this local picture, it would be reasonable to believe that the Earth is an inﬁnite plane, a sphere, a torus, or any other 2-manifold.

What other 2-manifold possibilities are there, and how can we eliminate the other possibilities?

Stacy Hoehn Manifolds and the Shape of the Universe Yes! There are actually inﬁnitely many compact 2-manifolds. But fortunately they are all made out of simple building blocks.

Compact 2-Manifolds

We will restrict our attention to 2-manifolds that are ﬁnite. These are called compact 2-manifolds.

The sphere and torus are both compact 2-manifolds. Are there any others?

Stacy Hoehn Manifolds and the Shape of the Universe Compact 2-Manifolds

We will restrict our attention to 2-manifolds that are ﬁnite. These are called compact 2-manifolds.

The sphere and torus are both compact 2-manifolds. Are there any others?

Yes! There are actually inﬁnitely many compact 2-manifolds. But fortunately they are all made out of simple building blocks.

Stacy Hoehn Manifolds and the Shape of the Universe This square, with its opposite sides identiﬁed, helps us depict the torus in the plane.

The Torus

To help us visualize the other compact 2-manifolds, we will ﬁrst view the torus a little bit diﬀerently. We will construct a torus by gluing together opposite edges of a square.

Stacy Hoehn Manifolds and the Shape of the Universe The Torus

To help us visualize the other compact 2-manifolds, we will ﬁrst view the torus a little bit diﬀerently. We will construct a torus by gluing together opposite edges of a square.

This square, with its opposite sides identiﬁed, helps us depict the torus in the plane.

Stacy Hoehn Manifolds and the Shape of the Universe You would see inﬁnitely many copies of yourself in every direction!

The Torus (continued)

What would you see if you were a two-dimensional being living in a torus?

Stacy Hoehn Manifolds and the Shape of the Universe The Torus (continued)

What would you see if you were a two-dimensional being living in a torus?

You would see inﬁnitely many copies of yourself in every direction!

Stacy Hoehn Manifolds and the Shape of the Universe The M¨obiusBand

A M¨obiusband is constructed from a square by gluing the left side to the right side of the square after performing a half-twist.

Stacy Hoehn Manifolds and the Shape of the Universe Note: The M¨obiusband is not a manifold because it has an edge. (It is called a manifold-with-boundary.)

The M¨obiusBand (continued)

A M¨obiusband contains an orientation-reversing curve. Clockwise becomes counterclockwise along this curve!

Stacy Hoehn Manifolds and the Shape of the Universe The M¨obiusBand (continued)

A M¨obiusband contains an orientation-reversing curve. Clockwise becomes counterclockwise along this curve!

Note: The M¨obiusband is not a manifold because it has an edge. (It is called a manifold-with-boundary.)

Stacy Hoehn Manifolds and the Shape of the Universe The Klein bottle is a 2-manifold.

The Klein Bottle

A Klein bottle is constructed from a square by gluing together the left and right edges the same way as for a torus, but now the top edge is ﬂipped before being glued to the bottom edge.

Stacy Hoehn Manifolds and the Shape of the Universe The Klein Bottle

A Klein bottle is constructed from a square by gluing together the left and right edges the same way as for a torus, but now the top edge is ﬂipped before being glued to the bottom edge.

The Klein bottle is a 2-manifold.

Stacy Hoehn Manifolds and the Shape of the Universe You would see inﬁnitely many copies of yourself in every direction, but sometimes you would be ﬂipped!

The Klein Bottle (continued)

What would you see if you were a two-dimensional being living in a Klein bottle?

Stacy Hoehn Manifolds and the Shape of the Universe The Klein Bottle (continued)

What would you see if you were a two-dimensional being living in a Klein bottle?

You would see inﬁnitely many copies of yourself in every direction, but sometimes you would be ﬂipped!

Stacy Hoehn Manifolds and the Shape of the Universe The Klein Bottle (continued)

The Klein bottle contains an orientation-reversing curve since it contains a M¨obiusband.

Manifolds that contain an orientation-reversing curve are called nonorientable. Manifolds that do not contain an orientation-reversing curve are called orientable.

Stacy Hoehn Manifolds and the Shape of the Universe The projective plane is a nonorientable 2-manifold that can be obtained from a disk by making diﬀerent identiﬁcations.

The Sphere and Projective Plane

The sphere is an orientable 2-manifold that can be obtained from a disk as shown below.

Stacy Hoehn Manifolds and the Shape of the Universe The Sphere and Projective Plane

The sphere is an orientable 2-manifold that can be obtained from a disk as shown below.

The projective plane is a nonorientable 2-manifold that can be obtained from a disk by making diﬀerent identiﬁcations.

Stacy Hoehn Manifolds and the Shape of the Universe Amazingly, every compact 2-manifold is homeomorphic to either a sphere (orientable), a connected sum of tori (orientable), or a connected sum of projective planes (nonorientable).

Connected Sum

Given two 2-manifolds, we can create a new 2-manifold by taking their connected sum.

To take the connected sum of two 2-manifolds, remove the inside of a small disk from each of them and then glue the two boundary circles of these disks together.

Stacy Hoehn Manifolds and the Shape of the Universe Connected Sum

Given two 2-manifolds, we can create a new 2-manifold by taking their connected sum.

To take the connected sum of two 2-manifolds, remove the inside of a small disk from each of them and then glue the two boundary circles of these disks together.

Amazingly, every compact 2-manifold is homeomorphic to either a sphere (orientable), a connected sum of tori (orientable), or a connected sum of projective planes (nonorientable).

Stacy Hoehn Manifolds and the Shape of the Universe There are inﬁnitely many 3-manifolds. A priori, any one of these 3-manifolds could be the shape of the universe.

The Shape of the Universe

No matter where we have been in the universe so far, if we choose a spot and travel out from it a short distance in all directions, we enclose a space that resembles a ball in 3-dimensional Euclidean space. Thus, the universe appears to be some 3-manifold. But which 3-manifold is it?

Stacy Hoehn Manifolds and the Shape of the Universe The Shape of the Universe

There are inﬁnitely many 3-manifolds. A priori, any one of these 3-manifolds could be the shape of the universe.

Stacy Hoehn Manifolds and the Shape of the Universe This limits the geometries (notions of distance, angles, and curvature) that can be placed on the universe’s 3-manifold to the following: spherical geometry with positive curvature Euclidean geometry with zero curvature hyperbolic geometry with negative curvature.

Narrowing Down the Possibilities

Scientists have measured the amount of cosmic microwave background radiation in the universe, and they have found that it is distributed surprisingly uniformly. This suggests that the curvature of the universe does not vary with either position or direction.

Stacy Hoehn Manifolds and the Shape of the Universe Narrowing Down the Possibilities

This limits the geometries (notions of distance, angles, and curvature) that can be placed on the universe’s 3-manifold to the following: spherical geometry with positive curvature Euclidean geometry with zero curvature hyperbolic geometry with negative curvature.

Stacy Hoehn Manifolds and the Shape of the Universe Curvature

In Euclidean geometry, the sum of the angles in a triangle is 180 degrees. Meanwhile, in spherical geometry, the sum of the angles is more than 180 degrees, and in hyperbolic geometry, the sum of angles is less than 180 degrees.

Stacy Hoehn Manifolds and the Shape of the Universe Euclidean Geometry ⇒ The universe will continue to expand forever, but just barely (i.e. the rate of expansion will approach 0.)

Hyperbolic Geometry ⇒ The universe will continue to expand forever, gradually approaching a (positive) constant rate of expansion.

Geometry and the Eventual Fate of the Universe

Spherical Geometry ⇒ The universe will eventually recollapse.

Stacy Hoehn Manifolds and the Shape of the Universe Hyperbolic Geometry ⇒ The universe will continue to expand forever, gradually approaching a (positive) constant rate of expansion.

Geometry and the Eventual Fate of the Universe

Spherical Geometry ⇒ The universe will eventually recollapse.

Euclidean Geometry ⇒ The universe will continue to expand forever, but just barely (i.e. the rate of expansion will approach 0.)

Stacy Hoehn Manifolds and the Shape of the Universe Geometry and the Eventual Fate of the Universe

Spherical Geometry ⇒ The universe will eventually recollapse.

Euclidean Geometry ⇒ The universe will continue to expand forever, but just barely (i.e. the rate of expansion will approach 0.)

Hyperbolic Geometry ⇒ The universe will continue to expand forever, gradually approaching a (positive) constant rate of expansion.

Stacy Hoehn Manifolds and the Shape of the Universe Data from a NASA probe in 2001 suggests that the curvature of the universe is very close to 0. This either means that we live in a Euclidean universe or we live in a spherical or hyperbolic universe with extremely low curvature.

Is the Universe Euclidean?

In the early 1800s, Carl Gauss computed the angles formed by 3 mountain peaks in Germany found that they added up to 180 degrees. However, this does not necessarily imply that the universe is Euclidean.

Stacy Hoehn Manifolds and the Shape of the Universe Is the Universe Euclidean?

Data from a NASA probe in 2001 suggests that the curvature of the universe is very close to 0. This either means that we live in a Euclidean universe or we live in a spherical or hyperbolic universe with extremely low curvature.

Stacy Hoehn Manifolds and the Shape of the Universe Yes! It narrows the number of possibilities down from inﬁnity to 18!

Theorem There are exactly 18 Euclidean 3-manifolds. 6 are compact and orientable 4 are compact and nonorientable 4 are noncompact and orientable 4 are noncompact and nonorientable

Euclidean 3-Manifolds

If we assume that the universe is a Euclidean 3-manifold, does this help us determine which manifold the universe is?

Stacy Hoehn Manifolds and the Shape of the Universe Theorem There are exactly 18 Euclidean 3-manifolds. 6 are compact and orientable 4 are compact and nonorientable 4 are noncompact and orientable 4 are noncompact and nonorientable

Euclidean 3-Manifolds

If we assume that the universe is a Euclidean 3-manifold, does this help us determine which manifold the universe is?

Yes! It narrows the number of possibilities down from inﬁnity to 18!

Stacy Hoehn Manifolds and the Shape of the Universe Euclidean 3-Manifolds

If we assume that the universe is a Euclidean 3-manifold, does this help us determine which manifold the universe is?

Yes! It narrows the number of possibilities down from inﬁnity to 18!

Theorem There are exactly 18 Euclidean 3-manifolds. 6 are compact and orientable 4 are compact and nonorientable 4 are noncompact and orientable 4 are noncompact and nonorientable

Stacy Hoehn Manifolds and the Shape of the Universe Nonorientable Euclidean 3-Manifolds

The 8 nonorientable Euclidean 3-manifolds all contain an orientation-reversing loop. If you were to ﬂy from Earth along such a loop, you would eventually return home with your orientation reversed. It would appear that you had returned to a mirror image of Earth.

If the universe was nonorientable, cosmologists predict that we would observe high amounts of energy radiating from regions where matter and anti-matter meet. While this could be happening outside of our ﬁeld of vision, they believe that it is unlikely that our universe is nonorientable.

Stacy Hoehn Manifolds and the Shape of the Universe The 3-Torus

The simplest orientable, compact, Euclidean 3-manifold is the 3-torus. It is a generalization of the torus in a higher dimension.

Instead of gluing together opposite edges of a square, the opposite faces of a cube are joined.

Stacy Hoehn Manifolds and the Shape of the Universe If the universe is a 3-torus, you could ﬂy from Earth in a particular direction and, without ever changing course, eventually return home.

The 3-Torus (continued)

If you were somehow in the 3-torus and looked around, you would see copies of yourself in each direction, and past these copies, other copies would be visible as far as the eye could see.

Stacy Hoehn Manifolds and the Shape of the Universe The 3-Torus (continued)

If you were somehow in the 3-torus and looked around, you would see copies of yourself in each direction, and past these copies, other copies would be visible as far as the eye could see.

If the universe is a 3-torus, you could ﬂy from Earth in a particular direction and, without ever changing course, eventually return home.

Stacy Hoehn Manifolds and the Shape of the Universe If you were inside the cube for the quarter-twist manifold and stared out the front or back face, you would see copy after copy of yourself, each one a 90-degree rotation of the preceding copy.

The Quarter-Twist and Half-Twist 3-Manifolds

In the quarter-twist and half-twist 3-manifolds, four of the faces of the cube are glued together just as for the 3-torus. The front and back faces, however, are glued together after a twist of 90 degrees (quarter-twist) or 180 degrees (half-twist).

Stacy Hoehn Manifolds and the Shape of the Universe The Quarter-Twist and Half-Twist 3-Manifolds

If you were inside the cube for the quarter-twist manifold and stared out the front or back face, you would see copy after copy of yourself, each one a 90-degree rotation of the preceding copy.

Stacy Hoehn Manifolds and the Shape of the Universe If you looked out of one of the hexagonal faces of the prism for the sixth-twist manifold, you would see copy after copy of yourself, each rotated 60 degrees more than the preceding copy.

The Sixth-Twist and Third-Twist 3-Manifolds

The sixth-twist and third-twist 3-manifolds are both obtained by gluing faces on a hexagonal prism instead of a cube. Each parallelogram face is glued to the face directly opposite it. The two hexagonal faces are then glued together after a twist of 60 degrees (sixth-twist) or 120 degrees (third-twist).

Stacy Hoehn Manifolds and the Shape of the Universe The Sixth-Twist and Third-Twist 3-Manifolds

If you looked out of one of the hexagonal faces of the prism for the sixth-twist manifold, you would see copy after copy of yourself, each rotated 60 degrees more than the preceding copy.

Stacy Hoehn Manifolds and the Shape of the Universe The Double Cube 3-Manifold

The last compact, orientable, Euclidean 3-manifold is the Double Cube manifold. It is important to note that not all of the faces for this manifold are glued to the ones across from them.

You would see yourself with a very peculiar perspective in this 3-manifold!

Stacy Hoehn Manifolds and the Shape of the Universe 3 The simplest one of these is 3-dimensional Euclidean space, R .

The others are called the Slab Space, the Chimney Space, and the Twisted Chimney Space.

Many cosmologists believe that the universe is not inﬁnite in nature, but we still must consider these 4 non-compact options as possibilities until there is substantial evidence against them.

Non-Compact, Orientable, Euclidean 3-Manifolds

It is likely that the universe has the shape of one of the six compact, orientable, Euclidean 3-manifolds that we just described. However, there are also 4 non-compact, orientable, Euclidean 3-manifolds.

Stacy Hoehn Manifolds and the Shape of the Universe The others are called the Slab Space, the Chimney Space, and the Twisted Chimney Space.

Many cosmologists believe that the universe is not inﬁnite in nature, but we still must consider these 4 non-compact options as possibilities until there is substantial evidence against them.

Non-Compact, Orientable, Euclidean 3-Manifolds

3 The simplest one of these is 3-dimensional Euclidean space, R .

Stacy Hoehn Manifolds and the Shape of the Universe Many cosmologists believe that the universe is not inﬁnite in nature, but we still must consider these 4 non-compact options as possibilities until there is substantial evidence against them.

Non-Compact, Orientable, Euclidean 3-Manifolds

3 The simplest one of these is 3-dimensional Euclidean space, R .

The others are called the Slab Space, the Chimney Space, and the Twisted Chimney Space.

Stacy Hoehn Manifolds and the Shape of the Universe Non-Compact, Orientable, Euclidean 3-Manifolds

3 The simplest one of these is 3-dimensional Euclidean space, R .

The others are called the Slab Space, the Chimney Space, and the Twisted Chimney Space.

Many cosmologists believe that the universe is not inﬁnite in nature, but we still must consider these 4 non-compact options as possibilities until there is substantial evidence against them.

Stacy Hoehn Manifolds and the Shape of the Universe Possible Problems: The gluing diagram for the universe is huge (possibly bigger than our sphere of vision) and is continuing to expand. Light travels at a finite speed, so looking out into the universe, we are looking back in time. Even if we someday find a copy of our galaxy, we may not recognize it because it might have looked different in its younger years.

Can We Narrow Down the Possibilities Even Further?

The simplest procedure is to look for copies of our galaxy, the Milky Way, in the night sky. If we ﬁnd copies, we can look at their pattern to determine the gluing diagram for the universe.

Stacy Hoehn Manifolds and the Shape of the Universe Can We Narrow Down the Possibilities Even Further?

The simplest procedure is to look for copies of our galaxy, the Milky Way, in the night sky. If we ﬁnd copies, we can look at their pattern to determine the gluing diagram for the universe.

Possible Problems: The gluing diagram for the universe is huge (possibly bigger than our sphere of vision) and is continuing to expand. Light travels at a finite speed, so looking out into the universe, we are looking back in time. Even if we someday find a copy of our galaxy, we may not recognize it because it might have looked different in its younger years.

Stacy Hoehn Manifolds and the Shape of the Universe More Information about the Shape of the Universe

Adams, Colin, and Robert Franzosa. Introduction to Topology: Pure and Applied. Upper Saddle River: Prentice Hall, 2007. Adams, Colin, and Joey Shapiro. “The Shape of the Universe: Ten Possibilities.” American Scientist. 89 (2001), no. 5, 443-453. Weeks, Jeﬀrey. The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds. New York: Marcel Dekker, Inc., 1985.

Stacy Hoehn Manifolds and the Shape of the Universe