Search results
Results from the WOW.Com Content Network
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 ...
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 ...
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 ...
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
Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology.
This is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary). If M {\displaystyle M} is oriented, analogously (i.e. counting intersections with signs) one defines the intersection form on the 2 {\displaystyle 2} nd homology group
A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an extraordinary homology theory (dually, extraordinary cohomology theory). Important examples of these were found in the 1950s, such as topological K-theory and cobordism theory , which are extraordinary co homology theories, and ...
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.