Simply connected topological spaces

Depends on

• Paths, path-connected spaces and path components
• Loop classes form the group $\pi_1(X,x_0)$ under concatenation

Used by

• A connected covering of a locally path-connected simply connected space is one-sheeted and trivial Corollary
• A simply connected overlap turns the van Kampen pushout into a free product Corollary
• Bonnet-Myers fundamental group is finite Corollary
• Finiteness of the fundamental group is sufficient, but not necessary, for Reeb stability Corollary
• Homotopy spheres of dimension at least five bounding a contractible manifold are standard spheres Corollary
• If one set in a van Kampen cover is simply connected, the other fundamental group surjects with overlap-generated kernel Corollary
• On a simply connected domain, pathwise continuation glues to one holomorphic function Corollary
• ℝ/ℤ is not simply connected Corollary
• The genus of a compact Riemann surface determines its uniformization type Corollary
• A four-dimensional boundary case is outside the smooth h-cobordism theorem Counterexample
• A nontrivial Whitney circle in the fundamental group blocks cancellation Counterexample
• Spherical, parabolic and hyperbolic universal-covering types Definition
• Universal covering spaces Definition
• Vanishing cycles of a codimension-one foliation Definition
• A complete locally CAT(0) circle whose fundamental group prevents global CAT(0) Example
• A complex torus has a lattice of parabolic deck translations Example
• A flat torus showing simple connectedness is needed for global exp injectivity Example
• A genus-two compact surface gives a cocompact Fuchsian group Example
• Annulus and punctured disc have hyperbolic universal covers Example
• Cartan hadamard for hyperbolic space Example
• Oppositely signed intersections of two three-manifolds in a simply connected six-manifold Example
• Simply connected h-cobordisms have zero Whitehead obstruction Example
• The claim that the ball and the polydisc are biholomorphic False statement
• A compact leafwise nullhomotopy persists under a transverse deformation Lemma
• A contractible relative group-ring complex with a pi-one isomorphism detects a homotopy equivalence Lemma
• A fixed leafwise cap gives a joint transverse product with exact collar data Lemma
• A simply connected Greenian Riemann surface is a disc Lemma
• A simply connected surface without a Green kernel is plane or sphere Lemma
• A universal covering of a Riemann surface inherits a unique complex structure Lemma
• Antipodal complements cover Sⁿ by simply connected sets with path-connected overlap for n≥2 Lemma
• Continuous characters of the real line are exponentials Lemma
• Finite wedges of quotient circles have van Kampen covers at the wedge point Lemma
• Harmonic conjugates and integral logarithmic-pole monodromy on surfaces Lemma
• Homology lemma: a handle-basis class is realized by a sphere meeting the belt sphere once Lemma
• Real Stiefel spaces with complement rank at least two are simply connected Lemma
• Removing the point at infinity preserves the knot group Lemma
• The cut surface, primitives of closed forms, and their boundary jumps Lemma
• The fundamental-group label controls contractibility of the Whitney circle Lemma
• The space of local geodesics, its length metric, and the covering criterion for local isometries Lemma
• The sphere, plane and disc are pairwise biholomorphically distinct Lemma

…and 22 more results.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• A. Hatcher, Algebraic Topology, Chapter 1, Proposition 1.6 (standard reference, not scraped)