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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.
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 ...
It also tells the same story. We have the honest Betti numbers 1, 1, 0. of the circle, and therefore 0, 1, 1. by flipping over and 1, 1, 0. by shifting to the left. This gives back something different from what the Jordan theorem states, which is that there are two components, each contractible (Schoenflies theorem, to be accurate about what is ...
There are two basic invariants of a space X in the rational homotopy category: the rational cohomology ring (,) and the homotopy Lie algebra ().The rational cohomology is a graded-commutative algebra over , and the homotopy groups form a graded Lie algebra via the Whitehead product.
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 ...
The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes: b 0 is the number of connected components; b 1 is the number of one-dimensional or "circular" holes; b 2 is the number of two-dimensional "voids" or "cavities".
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 ...