Civil Rights
Movements, leaders, victories and the continuing fight for equality.
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Movements, leaders, victories and the continuing fight for equality.
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Meet the people whose lives, choices and achievements shaped the journey.
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In August 1908, a white mob attacked Springfield, Illinois’s Black community, destroying homes and businesses and lynching two Black men. National outrage over the violence helped spur the movement that created the NAACP the following year.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups as the n-sphere, and so every homotopy sphere is necessarily a homology sphere.[1]
The topological generalized Poincaré conjecture is that any n-dimensional homotopy sphere is homeomorphic to the n-sphere; it was solved by Stephen Smale in dimensions five and higher, by Michael Freedman in dimension 4, and for dimension 3 (the original Poincaré conjecture) by Grigori Perelman in 2005.
In the case of smooth manifolds, in certain dimensions there exist homotopy spheres that are not diffeomorphic to the sphere, called exotic spheres. The smooth Poincaré conjecture, whether such non-trivial smooth homotopy spheres exist, is open in dimension 4.
Homotopy spheres form an abelian group known as Kervaire–Milnor group. Its composition is the connected sum and its neutral element is the sphere, while inversion is given by opposite orientation.
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In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups as the n-sphere, and so every homotopy sphere is necessarily a homology sphere. The topological generalized Poincaré conjecture is that any n-dimensional homotopy sphere is homeomorphic to the n-sphere; it was solved by Stephen Smale in dimensions five and higher, by Michael Freedman in dimension 4, and for dimension 3 (the original Poincaré conjecture) by Grigori Perelman in 2005. In the case of smooth manifolds, in certain dimensions there exist homotopy spheres that are not diffeomorphic to the sphere, called exotic spheres. The smooth Poincaré conjecture, whether such non-trivial smooth homotopy spheres exist, is open in dimension 4. Homotopy spheres form an abelian group known as Kervaire–Milnor group. Its composition is the connected sum and its neutral element is the sphere, while inversion is given by opposite orientation.
In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses and expansions (inverses of collapses), and a homotopy equivalence is a simple homotopy equivalence if it is homotopic to such a map. The obstruction to a homotopy equivalence being a simple homotopy equivalence is the Whitehead torsion, τ ( f ) . {\displaystyle \tau (f).} A homotopy theory that studies simple-homotopy types is called simple homotopy theory.
In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information about the point cloud or the distribution it is drawn from. Given a finite point cloud X and an ε > 0, we construct the Čech complex C ˇ ε ( X ) {\displaystyle {\check {C}}_{\varepsilon }(X)} as follows: Take the elements of X as the vertex set of C ˇ ε ( X ) {\displaystyle {\check {C}}_{\varepsilon }(X)} . Then, for each σ ⊂ X {\displaystyle \sigma \subset X} , let σ ∈ C ˇ ε ( X ) {\displaystyle \sigma \in {\check {C}}_{\varepsilon }(X)} if the set of ε-balls centered at points of σ has a nonempty intersection. In other words, the Čech complex is the nerve of the set of ε-balls centered at points of X. By the nerve lemma, the Čech complex is homotopy equivalent to the union of the balls, also known as the offset filtration.
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X {\displaystyle X} is denoted by π 1 ( X ) {\displaystyle \pi _{1}(X)} .
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Mae Jemison, aboard Space Shuttle Endeavour in 1992.
