# Does every topological manifold have a smooth structure?

Does every topological manifold have a smooth structure? It is well known that not every topological 4-manifold admits a smooth structure. Is every manifold smooth? Every smooth manifold is a topological manifold, but not necessarily

## Does every topological manifold have a smooth structure?

It is well known that not every topological 4-manifold admits a smooth structure.

## Is every manifold smooth?

Every smooth manifold is a topological manifold, but not necessarily vice versa. (The first nonsmooth topological manifold occurs in four dimensions.) Milnor (1956) showed that a seven-dimensional hypersphere can be made into a smooth manifold in 28 ways. Bredon, G. E. Topology & Geometry.

## Are manifolds topological spaces?

Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition.

## What is a smooth structure on a manifold?

Definition. A smooth structure on a manifold is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold is an atlas for such that each transition function is a smooth map, and two smooth atlases for.

## What is not a manifold?

Non-manifold geometry is defined as any edge shared by more than two faces. This can occur when a face or edge is extruded but not moved, which results in two identical edges directly on top of one another.

## Is a cone a smooth manifold?

The single cone can be very easily be considered a smooth manifold.

## Is the real line a manifold?

The real line is trivially a topological manifold of dimension 1. Up to homeomorphism, it is one of only two different connected 1-manifolds without boundary, the other being the circle. It also has a standard differentiable structure on it, making it a differentiable manifold.

## Is a cone a topological manifold?

Cone is a topological manifold.

## What are the different types of nail abnormalities?

Nail Abnormalities 1 Healthy nails appear smooth and have consistent coloring. 2 Abnormalities of the fingernail. Some changes in your nails are due to medical conditions… 3 Beau’s lines. Depressions that run across your fingernail are called Beau’s lines. 4 Clubbing. Clubbing is when your nails thicken and curve around your…

## Is it normal to have vertical ridges on your nails?

As you age, you may develop vertical ridges, or your nails may be a bit more brittle. This is harmless. Spots due to injury should grow out with the nail.