What is the connected sum of a sphere and a torus?
Connected sum with a torus is equivalent to adding a handle. Every surface is the connected sum of either the sphere, projective plane, or Klein bottle with zero or more tori.
What is the connected sum of two spheres?
Connected sum at a point A connected sum of two m-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation.
What surface do you get when you glue two Möbius bands together along their boundary?
Recall that a Möbius strip has only one boundary component. It is a circle. If you attach two Möbius strips together using a homeomorphism of the boundary circles, then the resulting surface is a non-orientable surface without boundary, and this is indeed a Klein bottle.
Is a projective plane orientable?
The projective plane is non-orientable.
How do you connect sums?
To connect sum two surfaces you pull out a disc from each, creating “holes”, and then sew the two surfaces together along the boundaries of the holes. This gives another surface!
Is connected sum associative?
Up to a diffeomorphism, the operation of taking connected sums is associative and commutative.
What is the relationship between a Klein bottle and a Mobius strip?
Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. Unlike the Möbius strip, the Klein bottle is a closed manifold, meaning it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot.
How many sides does a Klein bottle have?
one-sided
But a Klein Bottle does not have an edge. It’s boundary-free, and an ant can walk along the entire surface without ever crossing an edge. This is true of both theoretical Klein Bottles and our glass ones. And so, a Klein Bottle is one-sided.
Why is a torus orientable?
Orientable surfaces are surfaces for which we can define ‘clockwise’ consistently: thus, the cylinder, sphere and torus are orientable surfaces. In fact, any two-sided surface in space is orientable: thus the disc, cylinder, sphere and n-fold torus, all with or without holes, are orientable surfaces.
Is connected sum commutative?
To connect sum two surfaces you pull out a disc from each, creating “holes”, and then sew the two surfaces together along the boundaries of the holes. The operation is commutative, associative and there is even an identity element: just add a sphere to any surface and you get back that surface!
What is a Klein bottle in mathematics?
In topology, a branch of mathematics, the Klein bottle (/ˈklaɪn/) is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. While a Möbius strip is a surface with boundary, a Klein bottle has no boundary.
What is connected sum in geometry?
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces.
How to define the connected sum for two oriented knots?
To define the connected sum for two oriented knots: Consider a planar projection of each knot and suppose these projections are disjoint.
When is the connected sum of two manifolds unique?
If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth category, and then the result is unique up to diffeomorphism.