Statement
More accurately, we suppose given CW complexes X and Y, with respective base points x and y. Given a continuous mapping
such that f(x) = y, we consider for n ≥ 0 the induced homomorphisms
where πn denotes for n ≥ 1 the n-th homotopy group. For n = 0 this means the mapping of the path-connected components; if we assume both X and Y are connected we can ignore this as containing no information. We say that f is a weak homotopy equivalence if the homomorphisms f* are all isomorphisms. The Whitehead theorem then states that a weak homotopy equivalence, for connected CW complexes, is a homotopy equivalence.
Read more about this topic: Whitehead Theorem
Famous quotes containing the word statement:
“Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.”
—Charles Sanders Peirce (1839–1914)
“No statement about God is simply, literally true. God is far more than can be measured, described, defined in ordinary language, or pinned down to any particular happening.”
—David Jenkins (b. 1925)
“One is apt to be discouraged by the frequency with which Mr. Hardy has persuaded himself that a macabre subject is a poem in itself; that, if there be enough of death and the tomb in one’s theme, it needs no translation into art, the bold statement of it being sufficient.”
—Rebecca West (1892–1983)