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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.
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 ...
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.
To compute an extraordinary (co)homology theory for a CW complex, the Atiyah–Hirzebruch spectral sequence is the analogue of cellular homology. Some examples: For the sphere, S n , {\displaystyle S^{n},} take the cell decomposition with two cells: a single 0-cell and a single n -cell.
The 2025 CBSE board examination for Class 10 were held from 15 February till 18 March and from 15 February till 4 April for class 12. The usual starting time for each exam was 10:30 am( IST ) but depending on the length and/or maximum marks for the subject, the finishing time was either 12:30 pm ( IST ) (2 hours, shorter exams, usually 40-50 ...
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).
An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map h takes a homotopy class of maps from X to K(G, i) to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor. [1]