Homotopy : Homotopic 

Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H : X × [0,1] → Y from the product of the space X with the unit interval [0,1] to Y such that, for all points x in X, H(x,0)=f(x) and H(x,1)=g(x).

Being homotopic is an equivalence relation on the set of all continuous functions from X to Y. This homotopy relation is compatible with function composition in the following sense: if f₁, g₁ : X → Y are homotopic, and f₂, g₂ : Y → Z are homotopic, then their compositions f₂ o f₁ and g₂ o g₁ : X → Z are homotopic as well.

This allows to define the homotopy category: the objects are topological spaces, and the morphisms are homotopy classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy equivalent in the following sense: there exist continuous maps f : X → Y and g : Y → X such that g o f is homotopic to the identity map id_X and f o g is homotopic to id_Y. The maps f and g are called homotopy equivalences in this case.

Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, and R² - {(0,0)} is homotopy equivalent to the unit circle S¹. Those spaces that are homotopy equivalent to a point are called contractible.

Homotopy equivalence is important because in algebraic topology most concepts cannot distinguish homotopy equivalent spaces: if X and Y are homotopy equivalent, then

• if X is path-connected, then so is Y
• if X is locally path-connected, then so is Y
• if X is simply connected, then so is Y
• the homology and cohomology groups of X and Y are isomorphic
• if X and Y are path-connected, then the fundamental groups of X and Y are isomorphic, and so are the higher homotopy groups[?]

Especially in order to define the fundamental group, one needs the notion of homotopy relative to a subspace. These are homotopies which keep the elements of the subspace fixed. Formally: if f and g are continuous maps from X to Y and K is a subset of X, then we say that f and g are homotopic relative K if there exists a homotopy H : X × [0,1] → Y between f and g such that H(k,t) = f(k) for all k∈K and t∈[0,1].

Isotopy

In case the two given continuous functions f and g from the topological space X to the topological space Y are homeomorphisms, one can ask whether they can be connected 'through homeomorphisms'. This gives rise to the concept of isotopy, which is a homotopy H in the notation used before, such that for each fixed t, H(x,t) gives a homeomorphism.

In geometric topology - for example in knot theory - the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots K₁ and K₂ in three-dimensional space. The intuitive idea of deforming one to the other should correspond to a path of homeomorphisms: an isotopy starting with the identity homeomorphism of three-dimensional space, and ending at a homeomorphism h such that h moves K₁ to K₂.

I saw that diamond ring.html">ring.html">ring ain't no thief," went on Andy Royce. "I never stole anything in my once. I was mad at Miss Harrow for the way she treated me, an' just in. It was in the inkwell.html">inkwell that had red ink in it, an' the ring went hurry, an' left the seminary by the back door an' ran to the stables. me if there was any more trouble, so I knowed wot was comin'. Then I Sam's comment. "If this fellow's story.html">story is true.html">true.html">true, the ring ought to be in the inkwell for the summer.html">summer. In that case the person who cleaned the well ought to to believe.html">believe it or, not." "It's the truth!" cried Andy Royce. "We'll believe it when we see the ring," returned Tom, grimly. "I arrested," answered Tom. "But we are not going to let you get away know who is in charge.html">charge there during the summer?" "Why, I heard Nellie say that Miss Parsons took charge-- the teacher sure his story is true." Andy Royce demurred, but the boys would not listen to him. They belongings and pay his bill. Then, somewhat sobered by what was taking Here the boys hailed a passing taxicab that was empty, and ordered the the office, "but it may be true. People do queer things sometimes, had a grudge against Miss Harrow, and thought the disappearance of the for Nellie." It was arranged that Andy Royce should accompany Dick and Sam to the to Hope Seminary and received a.

 On wordlookup.net  

All is still licensed under the GNU FDL.
It uses material from the wikipedia.