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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 ...
[11] [12] 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.
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. [8]
The connectivity of the empty space is, by convention, conn H (X) = -2. Some computations become simpler if the connectivity is defined with an offset of 2, that is, ():= +. [2] The eta of the empty space is 0, which is its smallest possible value. The eta of any disconnected space is 1.
In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space X may be considered as dual to its complement in the n-sphere, where n is large enough. Its origins lie in Alexander duality theory, in homology theory, concerning complements in manifolds.
The Whitehead conjecture is equivalent to the conjecture that every sub-presentation of an aspherical presentation is aspherical. In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture , or there must be a counterexample to the Whitehead conjecture; in other words ...
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.
The cellular chain groups () are just freely generated on the -cells in degree , so they are in degree 0, and and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex , and since all boundary homomorphisms must be zero (recall that n > 1 {\displaystyle n>1} ), the cohomology is