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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 ...
This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers b i of X and the Betti numbers b i,F with coefficients in a field F. These can differ, but only when the characteristic of F is a prime number p for which there is some p-torsion ...
Alternatively, it can be constructed from two points x and y and two 1-dimensional balls A and B, such that the endpoints of A are glued to x and y, and the endpoints of B are glued to x and y too. A graph. Given a graph, a 1-dimensional CW complex can be constructed in which the 0-cells are the vertices and the 1-cells are the edges of the ...
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
Whitehead proved that simple homotopy equivalence is a finer invariant than homotopy equivalence by introducing an invariant called the torsion. The torsion of a homotopy equivalence takes values in a group now called the Whitehead group and denoted Wh(π), where π is the fundamental group of the target complex. Whitehead found examples of non ...
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.
Each scheme X over k determines two objects in DM called the motive of X, M(X), and the compactly supported motive of X, M c (X); the two are isomorphic if X is proper over k. One basic point of the derived category of motives is that the four types of motivic homology and motivic cohomology all arise as sets of morphisms in this category.
From the point of view of the definition, this is an expression of the fact that the knot complement is a homology circle, generated by the covering transformation . More generally if M {\displaystyle M} is a 3-manifold such that r a n k ( H 1 M ) = 1 {\displaystyle rank(H_{1}M)=1} it has an Alexander polynomial Δ M ( t ) {\displaystyle \Delta ...