Homotopy theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 31 May 2012

Maps and Based maps

Let X and Y be topological spaces.

HW: Show that Map(XY,Z) = Map(X,Map(Y,Z))

Let X and Y be based spaces.

HW: Show that Map*(XY,Z) = Map*(X, Map*(Y,Z)) .

Fundamental group, loop space and suspension

Let X be a based???? space.

HW: Show that Map*(SX,Y) Map*(X,ΩY), ΩnX= Map*(Sn,X), SnX =SnX, and [SnX,Y] =[X,ΩnY] .

The Eilenberg-Maclane spaces are important because the group homology and cohomology of G coincides with the homology and cohomology of K(G,1), Hn(G) Hn(K(G,1)) and Hn(G) Hn(K(G,1)) . Alternatively, K(G,1)BG.


Let φ:XX. A tricky way to view the fixed points of φ, Xφ = {xX | φ(x)=x} = {(x,x)X×X | φ(x)=x} = {(x1,x2) X×X | (x1, φ(x1)) =(x2,x2)} is as the push out Xφ X Δ X (id,φ) X×X where Δ(x)=(x,x) . The map Δ: X[0,1] X×X given by Δ(ω) =(ω(0), ω(1)), for ω:[0,1] X is homotopic to Δ.

Fibre bundles and classifying spaces

Example. Picture of Mobius band.

Let G be a group.

HW: If G is discrete BG =K(G,1) and EG is the universal cover of BG.

HW: EG is the unique, up to homotopy, contractible space on which G acts freely.


Let X and Y be topological spaces.

The Milnor construction of the classifying space of G is by letting G act on EG=G*G* ={ (t1g1, t2g2,) | ti[0,1], ti =1,most ti=0} {(t1g1, t2g2, ) = ( t1g1, t2g2, )if gi=gi for ti0} by (t1g1, t2g2,)g =(t1g1g, t2g2g,) and BG=EG/G.

Notes and References

These notes were written?????????


[Bou] N. Bourbaki, Algèbre, Chapitre ?: ??????????? MR?????.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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