Search results
Results from the WOW.Com Content Network
The Whitehead theorem states that a weak homotopy equivalence from one CW complex to another is a homotopy equivalence. (That is, the map f: X → Y has a homotopy inverse g: Y → X, which is not at all clear from the assumptions.) This implies the same conclusion for spaces X and Y that are homotopy equivalent to CW complexes.
If X is finite then Hom(X,Y) is homotopy equivalent to a CW complex by a theorem of John Milnor (1959). [14] Note that X and Y are compactly generated Hausdorff spaces, so Hom(X,Y) is often taken with the compactly generated variant of the compact-open topology; the above statements remain true. [15] Cellular approximation theorem
on homology is an isomorphism for all integers n. (Here H n (X) is the object of A defined as the kernel of X n → X n−1 modulo the image of X n+1 → X n.) The resulting homotopy category is called the derived category D(A).
Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing. Another important result is the approximation theorem. First, the homotopy category of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then
The elements of H n (X) are called homology classes. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous. [6] A chain complex is said to be exact if the image of the (n+1)th map is always equal to the kernel of the nth map.
The Whitehead group of a connected CW-complex or a manifold M is equal to the Whitehead group (()) of the fundamental group of M.. If G is a group, the Whitehead group is defined to be the cokernel of the map {} ([]) which sends (g, ±1) to the invertible (1,1)-matrix (±g).
The prototype here is the Jordan curve theorem, which topologically concerns the complement of a circle in the Riemann sphere. It also tells the same story. We have the honest Betti numbers 1, 1, 0. of the circle, and therefore 0, 1, 1. by flipping over and 1, 1, 0. by shifting to the left.
All India Secondary School Examination, commonly known as the class 10th board exam, is a centralized public examination that students in schools affiliated with the Central Board of Secondary Education, primarily in India but also in other Indian-patterned schools affiliated to the CBSE across the world, taken at the end of class 10. The board ...