Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Homotopy
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
=== Examples === * The first example of a homotopy equivalence is <math>\mathbb{R}^n</math> with a point, denoted <math>\mathbb{R}^n \simeq \{ 0\}</math>. The part that needs to be checked is the existence of a homotopy <math>H: I \times \mathbb{R}^n \to \mathbb{R}^n</math> between <math>\operatorname{id}_{\mathbb{R}^n}</math> and <math>p_0</math>, the projection of <math>\mathbb{R}^n</math> onto the origin. This can be described as <math>H(t,\cdot) = t\cdot p_0 + (1-t)\cdot\operatorname{id}_{\mathbb{R}^n}</math>. * There is a homotopy equivalence between <math>S^1</math> (the [[n-sphere|1-sphere]]) and <math>\mathbb{R}^2-\{0\}</math>. ** More generally, <math>\mathbb{R}^n-\{ 0\} \simeq S^{n-1}</math>. * Any [[fiber bundle]] <math>\pi: E \to B</math> with fibers <math>F_b</math> homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since <math>\pi:\mathbb{R}^n - \{0\} \to S^{n-1}</math> is a fiber bundle with fiber <math>\mathbb{R}_{>0}</math>. * Every [[vector bundle]] is a fiber bundle with a fiber homotopy equivalent to a point. * <math>\mathbb{R}^n - \mathbb{R}^k \simeq S^{n-k-1}</math> for any <math>0 \le k < n</math>, by writing <math>\mathbb{R}^n - \mathbb{R}^k</math> as the total space of the fiber bundle <math>\mathbb{R}^k \times (\mathbb{R}^{n-k}-\{0\})\to (\mathbb{R}^{n-k}-\{0\})</math>, then applying the homotopy equivalences above. * If a subcomplex <math>A</math> of a [[CW complex]] <math>X</math> is contractible, then the [[quotient space (topology)|quotient space]] <math>X/A</math> is homotopy equivalent to <math>X</math>.<ref>{{Cite book|title=Algebraic topology|last=Allen.|first=Hatcher|date=2002|publisher=Cambridge University Press|isbn=9780521795401|location=Cambridge|pages=11|oclc=45420394}}</ref> * A [[deformation retraction]] is a homotopy equivalence.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)