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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 ...
The short ladder in the complex solution in the 3, 2, 1 case appears to be tilted at 45 degrees, but actually slightly less with a tangent of 0.993. Other combinations of ladder lengths and crossover height have comparable complex solutions. With combination 105, 87, 35 the short ladder tangent is approximately 0.75.
Lastly, since there are many homology theories for topological spaces that produce the same answer, one also often speaks of the homology of a topological space. (This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in popular mathematics.)
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 .
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The Whitehead problem was the first purely algebraic problem to be proved undecidable. Shelah later showed that the Whitehead problem remains undecidable even if one assumes the continuum hypothesis. [3] [4] In fact, it remains undecidable even under the generalised continuum hypothesis. [5] The Whitehead conjecture is true if all sets are ...
It can be constructed from two points (x and y), and the 1-dimensional ball B (an interval), such that one endpoint of B is glued to x and the other is glued to y. The two points x and y are the 0-cells; the interior of B is the 1-cell. Alternatively, it can be constructed just from a single interval, with no 0-cells. A circle.
One usually makes the distinction between Whitehead's first and second lemma for the corresponding statements about first and second order cohomology, respectively, but there are similar statements pertaining to Lie algebra cohomology in arbitrary orders which are also attributed to Whitehead.