Classification of 2n-Manifolds Using Algebraic Methods
David Zhang
Advisor: Dr. James Turner
Manifolds
Classification Using Loops
Gluing Spheres
Higher Dimensions
The earth is a sphere, but it looks like a
plane when you stand on it. This is
because the earth is so large and our
field of vision is too small to capture the
curvature of the earth. This phenomena
is essential to the idea of a manifold: a
space in which if you “zoom in” at any
point, it will look like a point, a line, a
plane, or an Euclidean space of
dimension three or higher.
Although we can study the topological
structure of simple manifolds such as
balls and cubes by using our imagination
and common sense, the classification of
higher dimension manifolds requires
more systematic methods. One way of
doing this is by using a “contracting
loop”. If you draw a loop on a sphere or
a box, no matter what the loop looks like,
it can always contract to a point. On the
other hand, on a torus at least two kinds
of loops cannot contract to a point
without leaving the surface.
Another important fact about a highly
connected 2n-manifolds is that we can
obtain any of them by gluing a (2n-1)sphere to a bouquet of n-spheres. For the
two-dimensional case, a torus can be
obtained by gluing the boundary of a
disk to a skeleton made of circles. It
turns out that the structure of a 2nmanifold is determined by how we glue
it, which is described by its attaching
map. The attaching map can be
considered as a member of the (2n-1)-th
homotopy group of the bouquet of nsphere. Thus we can compare two 2nmanifolds by studying the associated
homotopy group elements.
The goal of our research is to develop an
algebraic method to classify 2nmanifolds. By using tools such as Pialgebra and Hilton Basis, we convert a
2n-manifold into an integer vector. The
overall stragety is shown in the
following flowchart:
For example, a circle is an 1-manifold
and a torus is a 2-manifold. Our universe
is a 3-manifold, but physicists have not
figured out whether it is a 3-Euclidean
space, a 3-sphere, or a 3-torus.
Structure of Manifolds
Manifolds come in all shapes and sizes.
However, in topology, the study of
geometric properties unaffected by
continuous changes, we are only
concerned with the structure of a
manifold. A ball and a cube look quite
different, but if the ball is made of PlayDoh, you can reshape the ball into a
cube without tearing or connecting any
parts. However, there is no way you can
reshape a ball into a doughnut without
poking a whole or connecting two ends.
This means that a ball is topologically
equivalent to a cube, while
topologically different from a torus.
The contractibility of loops on a surface
gives rise to an algebraic structure called
homotopy groups. Each manifold has a
unique homotopy group. It turns out that
two manifolds have the same structure
(of the same homotopy type) if and only
if their homotopy groups are the same.
Manifold
Cell Complex
Attaching Map
Homotopy Group Element
Vector
What we discovered is that two 2nmanifolds are of the same homotopy
type if and only if their associated
vectors can be mapped to each other by a
change-of-basis matrix that satisfies
certain properties. This necessary and
sufficient condition holds for manifolds
of any even dimensions.
Hopf fibration, a 3-dimensional
representation of a 3-sphere, which
resides in a 4-demsnional space.
References
A mug can be continuously transformed into a doughnut and
therefore the two objects are topologically equivalent
Any loop on a sphere can contract to a point, but this is
not true for loops on a torus
By gluing the edges of a square in different ways, we
can form a torus or a sphere.
Photo credits: Intelligent Perception (mug), Scientific
American (loops on torus), Wikipedia (loop on sphere, Hopf
fibration), Annenberg Learner (gluing a square).