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For instance, take X= S 2 × RP 3 and Y= RP 2 × S 3. Then X and Y have the same fundamental group, namely the cyclic group Z/2, and the same universal cover, namely S 2 × S 3; thus, they have isomorphic homotopy groups. On the other hand their homology groups are different (as can be seen from the Künneth formula); thus, X and Y are not ...
CW complexes satisfy the Whitehead theorem: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups. A covering space of a CW complex is also a CW complex. [13] The product of two CW complexes can be made into a CW complex.
That is, the correct answer in honest Betti numbers is 2, 0, 0. Once more, it is the reduced Betti numbers that work out. With those, we begin with 0, 1, 0. to finish with 1, 0, 0. From these two examples, therefore, Alexander's formulation can be inferred: reduced Betti numbers ~ are related in complements by
In homological algebra, Whitehead's lemmas (named after J. H. C. Whitehead) represent a series of statements regarding representation theory of finite-dimensional, semisimple Lie algebras in characteristic zero. Historically, they are regarded as leading to the discovery of Lie algebra cohomology. [1]
(For X and Y path-connected, the first condition is automatic, and it suffices to state the second condition for a single point x in X.) For simply connected topological spaces X and Y, a map f: X → Y is a weak homotopy equivalence if and only if the induced homomorphism f *: H n (X,Z) → H n (Y,Z) on singular homology groups is bijective ...
The Brouwer fixed point theorem: every continuous map from the unit n-disk to itself has a fixed point. The free rank of the n th homology group of a simplicial complex is the n th Betti number , which allows one to calculate the Euler–Poincaré characteristic .
Two pairs (X 1, A) and (X 2, A) are said to be equivalent, if there is a simple homotopy equivalence between X 1 and X 2 relative to A. The set of such equivalence classes form a group where the addition is given by taking union of X 1 and X 2 with common subspace A. This group is natural isomorphic to the Whitehead group Wh(A) of the CW-complex A.
For all integers r ≥ r 0, an object E r, called a sheet (as in a sheet of paper), or sometimes a page or a term, Endomorphisms d r : E r → E r satisfying d r o d r = 0, called boundary maps or differentials, Isomorphisms of E r+1 with H(E r), the homology of E r with respect to d r. The E 2 sheet of a cohomological spectral sequence