proposition 14.31 $\SO(3,\R)$ is not simply connected

open in the book · parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1189 · p. 555

Rests on

depends_on corollary 14.38 $\SU(2)$ is the universal cover ¶
- depends_on proposition 14.33 $\SU(2)$ is the three-sphere ¶
  - depends_on definition 14.32 The special unitary group in two dimensions ¶
  - proves proof ch:12-lie-groups-fibre-bundles@proof-11 ¶
- depends_on theorem 14.37 $\SU(2)$ is a two-to-one cover of $\SO(3,\R)$ ¶
  - depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶
    - depends_on equation 14.28 eq:lie-expso2 ¶
    - depends_on equation 14.25 eq:lie-so2-matrix ¶
    - depends_on proposition 14.27 Generators of $\mathfrak{so}(3)$ ¶
      - depends_on definition 14.26 The rotation group ¶
      - proves proof ch:12-lie-groups-fibre-bundles@proof-8 ¶
    - proves proof ch:12-lie-groups-fibre-bundles@proof-9 ¶
  - depends_on proposition 14.36 Closed form of the exponential ¶
    - depends_on equation 14.28 eq:lie-expso2 ¶ ↺
    - depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶
      - depends_on definition 14.32 The special unitary group in two dimensions ¶ ↺
      - proves proof ch:12-lie-groups-fibre-bundles@proof-13 ¶
    - proves proof ch:12-lie-groups-fibre-bundles@proof-14 ¶
  - depends_on proposition 14.35 Generators of $\mathfrak{su}(2)$ ¶ ↺
  - depends_on proposition 14.33 $\SU(2)$ is the three-sphere ¶ ↺
  - proves proof ch:12-lie-groups-fibre-bundles@proof-15 ¶
- proves proof app:A-long-proofs@proof-225 ¶
- proves proof ch:12-lie-groups-fibre-bundles@prooflink-4 ¶
depends_on definition 6.17 Simply connected space ¶
- depends_on definition 6.6 Continuous map ¶
  - depends_on definition 3.53 Preimage ¶
  - depends_on definition 6.2 Open set ¶
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - depends_on definition 6.1 Topological space ¶ ↺
- depends_on definition 6.15 Path-connected space ¶
  - depends_on definition 6.6 Continuous map ¶ ↺
depends_on proposition 14.30 Rodrigues formula; the exponential map is onto ¶ ↺
proves proof ch:12-lie-groups-fibre-bundles@proof-10 ¶

Supports

depends_on remark 29.12 The coordinate singularity, and what it is not ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	$\SU(2)$ is the universal cover	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1199
`depends_on`	→	Simply connected space	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1199
`depends_on`	→	Rodrigues formula; the exponential map is onto	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1199
`depends_on`	←	The coordinate singularity, and what it is not	declared	parts/03-classical-mechanics/12-rigid-body-rotating-frames.tex:345
`proves`	←	ch:12-lie-groups-fibre-bundles@proof-10	declared	parts/02-mathematical-methods/12-lie-groups-fibre-bundles.tex:1202