homotopy

The homotopy groups, however, carry information about the global structure.

They are dual to the homotopy groups, but less studied.

Subsequently the name homotopy Lie group has also been used.

Thus, both the first and second homotopy groups of a space are contained within its fundamental 2-group.

Then a homotopy between the two systems is considered.

This means that the rational homotopy of a formal space is particularly easy to work out.

The fundamental group is the first and simplest of the homotopy groups.

And if they were right, the table of homotopy groups started to look periodic for a long stretch.

Following this line of thought, an entire stable homotopy category can be created.

There is a long exact sequence of relative homotopy groups.

The history in relation to homotopy groups is interesting.

It is possible to reconstruct the free homotopy type of f from these data.

In some cases it can be shown that the higher homotopy groups of Y are trivial.

This can then be used to prove the commutativity of the higher homotopy groups.

Homotopy groups above do not change under covers, so they agree with those of the orthogonal group.

This can be formulated in terms of homotopy theory.

These are labeled by elements of the second homotopy group .

There is no continuous transformation that will map a solution in one homotopy class to another.

The double of a Mazur manifold is a homotopy 4-sphere.

The motion of the leash describes a homotopy between the two curves.

The homotopy must be a 1-parameter family of immersions.

When K is a point, the term pointed homotopy is used.

In this way, 2-groups classify pointed connected weak homotopy 2-types.

If all these vanish, then the map is a homotopy equivalence.

It is possible to define abstract homotopy groups for simplicial sets.