snnnnniiiiiiffffffffffff…oh yes my dear….sssnnnnnnnnnnnniiiiiiiiffffffff….quite pungent indeed…is that….dare I say….sssssssnniff…eggs I smell?......sniff sniff….hmmm…yes…quite so my darling….sniff….quite pungent eggs yes very much so …..ssssssssssssssnnnnnnnnnnnnnnniiiiiiiffffff….ah yes…and also….a hint of….sniff….cheese…..quite wet my dear….sniff…but of yes…this will do nicely….sniff…..please my dear….another if you please….nice a big now….BBBBBBRRRRRRRAAAAAAAPPPPPPPFFFFFFFFLLLLLLLLLPPPPPPPPPFFFFFF Oh yes…very good!....very sloppy and wet my dear….hmmmmm…is that a drop of nugget I see on the rim?...hmmmm…..let me…..let me just have a little taste before the sniff my darling…....hmmmmm….hmm..yes….that is a delicate bit of chocolate my dear….ah yes….let me guess…curry for dinner?....oh quite right I am….aren’t I?....ok….time for sniff…..sssssnnnnnnniiiiiiiiffffffff…..hmmm…hhhmmmmm I see…yes….yes indeed as well curry…...hmmm….that fragrance is quite noticeable….yes…..onion and garlic chutney I take it my dear?.....hmmmmm….yes quite…..BBBBBBRRRRRRRRPPPPPPFFFFFFFFFFFFFFFFFFFFFTTTTTTTTTTT Oh I was not expecting that…that little gust my dear….you caught me off guard…yes…so gentle it was though…hmmmm…let me taste this little one…just one small sniff…..sniff…ah….ssssssnnnnnniiiiiffffffffffff…and yet…so strong…yes…the odor….sniff sniff…hmmm….is that….sniff….hmmm….I can almost taste it my dear…..yes….just…sniff….a little whiff more if you please…..ssssssnnnnnniiiiiffffffffff…ah yes I have it now….yes quite….hhhhmmmm…delectable my dear…..quite exquisite yes…..I dare say…sniff….the most pungent one yet my dear….ssssnnnnniiiifffffffffffffffffffffff….yes….

Def: Let p : E → B be a map. If f : X → B is a map,

a lifting of f is a map

f : X → E such that p ◦

f = f

Diagrams: E

p

X f

f 

B

(E,e0)

p

(I,0) f

f



(B,b0)

Lemma: (Path Lifting) Let p : E → B be a covering map.

Let p(e0) = b0. Any path f : [0,1] → B beginning at b0 has

a unique lifting to a path

f in E beginning at e0.

Mth 632 – Winter 2009 Path and Homotopy Lifting 1/5

Homotopy Lifting

Lemma: (Homotopy Lifting) Let p : E → B be a covering

map. Let p(e0) = b0. Any homotopy H : [0,1]×[0,1] → B

with

H(0) = b0 has a unique lifting to a homotopy H

in E with

H(0) = e0. If H is a path homotopy, so is H

.

(E,e0)

p

(I ×I,0) H

H









 

(B,b0)

Mth 632 – Winter 2009 Path and Homotopy Lifting 2/5

Lifting Correspondence

Thm: Let p : E → B be a covering map. Let p(e0) = b0.

paths f and g from b0 to b1 are path homotopic iff

f and

g,

the lifts of f and g beginning at e0, end at the same point

and are path homotopic.

Def: Let p : E → B be a covering map. Let b0 ∈ B. Choose

an e0 so that p(e0) = b0. Given [f] ∈ π1(B,b0), let

f be the

lifting of f to a path in E beginning at e0. Let φe0(f) be

f(1).

φe0 : π1(B,b0) → p−1(b0)

is called the lifting correspondence derived from p.

Mth 632 – Winter 2009 Path and Homotopy Lifting 3/5

Fundamental G