Difference between revisions of "Nth homotopy group"
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{{DISPLAYTITLE:{{M|n^\text{th} }} homotopy group}} | {{DISPLAYTITLE:{{M|n^\text{th} }} homotopy group}} | ||
| + | : '''Note: ''' the [[fundamental group]] is {{M|\pi_1(X,p)}} | ||
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** {{M|f\circ\pi}} is [[continuous]] {{iff}} {{M|f}} is continuous | ** {{M|f\circ\pi}} is [[continuous]] {{iff}} {{M|f}} is continuous | ||
That gives us an association between continuous maps of the form {{M|f\circ\pi}} with some constraints. Blah blah blah, something like that. | That gives us an association between continuous maps of the form {{M|f\circ\pi}} with some constraints. Blah blah blah, something like that. | ||
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| + | [[Pointed topological space|Pointed topological spaces]] are involved. | ||
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==References== | ==References== | ||
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{{Definition|Topology|Algebraic Topology}} | {{Definition|Topology|Algebraic Topology}} | ||
Latest revision as of 22:17, 12 December 2016
- Note: the fundamental group is π1(X,p)
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Contents
[hide]Definition
Let (X,J) be a topological space with x0∈X being any fixed point. The nth homotopy group, written πn(X,x0) is defined as follows:
- The underlying set of the group is: πn(X,x0):=[(Sn,p),(X,x0)]∗
- Where [(Sn,p),(X,x0)]∗ denotes equivalence classes of continuous maps where f(p)=x0 under the equivalence relation of homotopy relative to p, i.e.:
- [(Sn,p),(X,x0)]∗:={f∈C(Sn,X) | f(p)=x0}((⋅) ≃ (⋅) (Rel {p}))
- Where [(Sn,p),(X,x0)]∗ denotes equivalence classes of continuous maps where f(p)=x0 under the equivalence relation of homotopy relative to p, i.e.:
Caveat:I think, the book... pointed spaces are really not that special, I'm using Books:Topology and Geometry - Glen E. Bredon for this
Noting that:
- (Sn,p)≅RS(Sn−1) where RS denotes the reduced suspension of a space we see:
- (Sn,p)≅RS(Sn−1,p):=(Sn−1×I({p}×I)∪(Sn−1×{0,1}),π(p,i)) for π the quotient map of some sort or other, where i∈I doesn't matter, as they're all the same under the quotient map.
- I:=[0,1]⊂R
- (Sn,p)≅RS(Sn−1,p):=(Sn−1×I({p}×I)∪(Sn−1×{0,1}),π(p,i)) for π the quotient map of some sort or other, where i∈I doesn't matter, as they're all the same under the quotient map.
Notes
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Although not the best quotient we do have the situation on the right:
- By the characteristic property of the quotient topology:
- f∘π is continuous if and only if f is continuous
That gives us an association between continuous maps of the form f∘π with some constraints. Blah blah blah, something like that.
Pointed topological spaces are involved.
