Ttykuu

I want to draw a diagram similar to this one:

For that I started with the following code:

begintikzpicture
draw (0,0) node $Y$;
draw (0,2) node $X$;
draw[<-] (0,0.35) -- (0,1.65) node[left, midway] $f$;
draw[thick] (1,0) -- (7,0);
draw[thick] (1,2) -- (7,2);
draw[thick] (1,2.5) -- (7,2.5);
draw[thick] (1,1.5) -- (7,1.5);
endtikzpicture

The only thing that I don't know how to do is the curvy parts. I would appreciate some indication.

The following is a pretty manual way to do this. I only did it for the first two lines, I hope you can apply it to the other occurrences. It uses the in and out keys of the to path construction:

documentclass[tikz]standalone

begindocument
begintikzpicture
draw (0,0) node $Y$;
draw (0,2) node $X$;
draw[<-] (0,0.35) -- (0,1.65) node[left, midway] $f$;
draw[thick] (1,2.5) -- (7,2.5) coordinate(a);
draw[thick] (1,2) -- (7,2) coordinate(b);
draw[thick] (1,1.5) -- (7,1.5) coordinate(c);
draw[thick] (1,0) -- (7,0) coordinate(d);
draw[thick]
 (a) ++(.25,-.25) coordinate(ab) to[out=180,in=0] (a)
 (ab) to[out=180,in=0] (b)
 (ab) to[out=0,in=180] ++(.25,.25)
 (ab) to[out=0,in=180] ++(.25,-.25)
 ;
filldraw
 (ab) circle(.05)
 ;
endtikzpicture
enddocument

This uses the same in and out trick as Skillmon and puts it into a style dip, which takes as arguments the horizontal position and the depth, where the sign decides whether the dip is a dip (minus) or a bump (plus).

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarypositioning
newcounterdip
begindocument
begintikzpicture[dip/.style args=#1/#2/utils/exec=stepcounterdip,
insert path=-aux1)
]
beginscope[thick,local bounding box=dips]
 draw (1,2.5) [dip=5.5cm/-2.5mm]-- (7,2.5);
 fill (dip-1) circle[radius=2pt] node[right=3pt]$b=1$;
 draw (1,2) [dip/.list=2.5cm/-2.5mm,5.5cm/2.5mm] -- (7,2);
 fill (dip-2) circle[radius=2pt] node[right=3pt]$b=1$;
 draw (1,1.5) [dip/.list=2.5cm/2.5mm,5.5cm/-5mm] -- (7,1.5);
 fill (dip-5) circle[radius=2pt] node[above right=0pt and 5pt]$b=2$;
 draw (1,1) -- (7,1);
 draw (1,0.5) [dip=5.5cm/5mm] -- (7,0.5);
endscope
path (dips.north west) node[anchor=north east] (X) $X$;
path (dips.south west) node[anchor=south east] (Y) $Y$;
draw[<-] (Y) -- (X) node[left, midway] $f$;
endtikzpicture
enddocument