Homotopy Theory

The fundamental group (\pi_1(X)) is defined as the group of all homotopy classes of continuous maps from the unit circle (S^1) to X, where two maps are homotopic if they can be continuously deformed into each other while keeping the endpoints fixed.

The homology group (H_n(X)) is defined as the group of all singular n-chains in X, modulo the group of all singular n-boundaries in X. A singular n-chain is a formal sum of n-simplices in X, and a singular n-boundary is a boundary of an (n+1)-simplex in X.

The cohomology group (H^n(X)) is defined as the group of all singular n-cocycles in X, modulo the group of all singular n-coboundaries in X. A singular n-cochain is a function that assigns an integer to each n-simplex in X, and a singular n-coboundary is the coboundary of an (n-1)-cochain.

The Hurewicz theorem states that for a simply connected space X, there is an isomorphism between the homology groups (H_n(X)) and the homotopy groups (\pi_n(X)) for (n \geq 2).

The Whitehead theorem states that if X and Y are two spaces and (f: X \to Y) is a homotopy equivalence, then (f) induces an isomorphism between the homology groups (H_n(X)) and (H_n(Y)) for all (n).

The Poincare duality theorem states that for a closed, oriented n-manifold M, there is an isomorphism between the homology group (H_n(M)) and the cohomology group (H^{n-n}(M)).

The homology suspension theorem states that for a space X, there is an isomorphism between the homology groups (H_n(X)) and the homology groups (H_{n+1}(\Sigma X)) for all (n).

The Eilenberg-Steenrod axioms are a set of axioms that characterize the homology and cohomology groups of a space. They provide a foundation for the study of homology and cohomology theory.

The Kunneth theorem states that for two spaces X and Y, there is an isomorphism between the homology groups (H_n(X \times Y)) and the direct sum (\bigoplus_{i+j=n} H_i(X) \otimes H_j(Y)) for all (n).

The Lefschetz duality theorem states that for a compact, oriented n-manifold M, there is an isomorphism between the homology group (H_n(M)) and the cohomology group (H^{n-n}(M)).

The Alexander duality theorem states that for a compact, connected, orientable 3-manifold M, there is an isomorphism between the homology group (H_1(M)) and the cohomology group (H^2(M)), and an isomorphism between the homology group (H_2(M)) and the cohomology group (H^1(M)).

A homology sphere is a sphere that is homology equivalent to the n-sphere (S^n). This means that there is an isomorphism between the homology groups of the sphere and the homology groups of (S^n).

The Poincare conjecture states that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. This conjecture was proven by Grigori Perelman in 2002.

The sphere theorem states that every homotopy sphere is homeomorphic to a sphere. This theorem was proven by Stephen Smale in 1961.

The Freudenthal suspension theorem states that the suspension of a pointed space is a simply connected space. This theorem was proven by Hans Freudenthal in 1937.