We construct a functor AC(−, −) from the category of path connected spaces X with a base point x to the category of simply connected spaces. The following are the main results of the paper: (i) If X is a Peano continuum then AC(X, x) is a cell-like Peano continuum; (ii) If X is n-dimensional then AC(X, x) is (n + 1)−dimensional; and (iii) For a path connected space X, π1(X, x) is trivial if and only if π2(AC(X, x)) is trivial. As a corollary, AC(S1, x) is a 2-dimensional nonaspherical cell-like Peano continuum.
Mathematics Subject Classification (2010)
Primary 54F15 55N15 Secondary 54G20 57M05
Noncontractible compactum weak homotopy equivalence trivial shape Peano continuum Snake cone Alternating cone asphericity cell-like space Topologist sine curve
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