1. FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
Petar Pavešić, University of Ljubljana
In fibrewise homotopy theory one often needs fibrewise versions of various functorial constructions on pointed topological spaces (e.g. loop-spaces, Pontryagin-Thom construction, etc.). We will present a general approach to this problem that is based on the construction of free topological groups.
Eilat 1

2. FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
1. INTRODUCTION
fibrewise space p: X B = family of fibres Xb=p-1(b) continuously
parametrized by points of B
fibrewise pointed space
B X B s p
ps=1B
= continuous family of pointed spaces (Xb, s(b))
To develop fibrewise (pointed) topology and homotopy theory we need fibrewise versions of standard pointed and unpointed operations on topological spaces, e.g. products, wedges, smashes, suspensions, loops, joins, localizations, Postnikov decompositions, Ganea constructions,… 2
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

3. Whitney sum (fibrewise product) XBY=(x,y) XY |p(x)=q(y)
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
Whitney sum (fibrewise product)
XBY=(x,y) XY |p(x)=q(y)
Husemoller: prolongation of a continuous functor
: (Vec)m (Vec*)n Vec
to a functor on vector bundles.
and other similar constructions like fibrewise joins etc.
James: fibrewise extension of a continuous functor : Top Top
Crabb-James: explicit construction of fibrewise pointed wedges, smashes and some other pointed operations.
May-Sigurdsson: explicit constructions and extension of limits and colimits in Top• to fibrewise limits and colimits in fibrewise pointed spaces.
Iwase-Sakai: extension of a continuous functor : Top• Top• to locally trivial fibrewise pointed spaces.
P. : fibrewise extension of a continuous functor : Top• Top• for total spaces that embed in products of . 3
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

4. FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
James: (fibrewise extension of a continuous functor : Top Top) endows with a suitable initial topology wrt a map to B (X).
 (Xb)
bB
Approach does not extend to functors on Top• because the inclusion map is not pointed.
Iwase-Sakai: carefully patches sets U(F) over a locally trivial fibrewise pointed space.
Unfortunately, many important fibrewise pointed spaces are not locally trivial: pr  c 
Diagonal and constant fibrewise pointed spaces are equivalent as fibrewise spaces but are not equivalent as fibrewise pointed spaces. 4
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

5. Fibrewise wedge of fibrewise pointed spaces and is given by
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
B X B s p r q
Fibrewise wedge of fibrewise pointed spaces
and
is given by
X B Y := {(x,y)X×Y | p(x)=q(y), x=sp(x) or y=rq(y)}
with obvious projection and section.
Both factors are fibrewise trivial but the fibres of the fibrewise wedge are not all homeomorphic, so the resulting fibrewise (pointed) space is not even locally trivial as a fibrewise space.
This difficulty arises in all non-homogeneous spaces.
P. : extension of : Top• Top• for total spaces that embed in products of .
Construction depends on the embedding (at least in principle) so it is hard to prove functoriality and naturality. Also, it is hard to extend to functors of more variables. 5
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

6. 2. CONSTRUCTION i X FX f G X : topological space
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
2. CONSTRUCTION
X : topological space X FX G i f
There exists a free topological group FX, such that
Underlying set FX is the free group generated by X.
Topology on FX is the maximal one for which i is an embedding.
(if there is one, take union of all topologies with that property – A.A.Markov 1941, …)
FACTS:
i:XFX is an embedding  X is a completely regular space
i:XFX is a closed embedding  X is a Tychonoff space
FX is Hausdorff  X is functionally Hausdorff
FX is a k-space  X is locally compact and separable or discrete 6
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

7.  (( Xk)b,sk (b))k=1,...,n  {b}× (FXk,ek )k=1,...,n
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
CReg• pointed completely regular spaces
CRegB fibrewise pointed completely regular spaces B
so that ib: (Xb,s(b))  (FX,e)
B X B s p
For in CRegB define ib: X  FX as ib(x) = i(x)i(s(b))-1 B
Given
Bk Xk Bk sk pk
in CRegB for k=1,...,n B
: (CReg•)n  Top• functor
and
Define (X1,..., Xn) to be the set  (( Xk)b,sk (b))k=1,...,n , endowed with the initial
topology, induced by the function
bB
 (( Xk)b,sk (b))k=1,...,n  {b}× (FXk,ek )k=1,...,n
bB
 ((ik)b) k b
 B × (FXk,ek )
p :=prB ° : (X)  B
s : B  (X) maps to fibre basepoints
 ((ik)b) k b 7
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

8. f *(X1,..., Xn)  (f *(X1),..., f *(Xn))
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
Theorem
(1) For every functor : (CReg•)n  Top• the correspondence
defines a fibrewise pointed extension functor
= B : (CRegB)n  TopB
(2) A natural transformation of functors
: 
induces a natural transformation of their fibrewise pointed extensions
B : B B
(3) Fibrewise pointed extension commutes with pull-backs: for f: A B
f *(X1,..., Xn)  (f *(X1),..., f *(Xn))
(Bk Xk Bk) k=1,...,n  (B (X1,..., Xn) B) sk pk p s B B B B B 8
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

9. (Bk Xk Bk) k=1,...,n (B (X1,..., Xn) B)
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
(4) If are locally fibrewise pointed trivial or sectioned
then so is
(5) If the functors ,: CReg•  CReg• form an adjoint pair, then so do
their fibrewise pointed extensions B,B : CRegB  CRegB
(Bk Xk Bk) k=1,...,n sk pk p s
(B (X1,..., Xn) B) B B B B
For standard functors (product, fat wedge, smash, join, loop,...) B
coincides with the constructions of James-Crabb, May-Sigurdsson,... B 9
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

10. (i1...in): ((X1,x1),..., (Xn,xn))  ((E1,x1),..., (En,xn)).
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
 preserves (closed) subspaces if for every (closed) subspaces ik :(Xk,xk)  (Ek,xk) the topology on ((X1,x1),..., (Xn,xn)) coincides with the topology induced by
(i1...in): ((X1,x1),..., (Xn,xn))  ((E1,x1),..., (En,xn)).
products, (fat) wedges, based path spaces, based loop spaces... preserve subspaces
one-point compactification does not preserve subspaces
smash products, joins,... preserve closed subspaces
Theorem
If the functor : (CReg•)n  Top• preserves subspaces then (X1,..., Xn) is a fibrewise pointed space whose fibres are ((X1)b,..., (Xn)b).
If the functor : (CReg•)n  Top• preserves closed subspaces and if X1,..., Xn are Tychonoff spaces then the fibres of (X1,..., Xn) over B
are ((X1)b,..., (Xn)b). 10
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

11. f : A(X1,..., Xn) (Y1,..., Yn), f (a,u):=( f (a, –))(u)
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
 is (topologically) continuous if for every collection of continuous maps fk: AXk Yk
the induced map
f : A(X1,..., Xn) (Y1,..., Yn), f (a,u):=( f (a, –))(u)
is also continuous.
products, wedges, path and loop spaces, smash products, joins,... are continuous
Theorem
If the functor : (CReg•)n  Top• is topologically continuous, and if
are Hurewicz fibrations then
is also a Hurewicz fibration.
(Bk Xk Bk) k=1,...,n sk pk p s
(B (X1,..., Xn) B) 11
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

12. FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
3. SOME APPLICATIONS
(a) GANEA – WHITEHEAD FRAMEWORK FOR TOPOLOGICAL COMPLEXITY
cat(X): Lusternik – Schnirelmann (LS) category of X is the size of the minimal
cover of X by open subsets that can be deformed to a point within X.
WnX
fat wedge
nX X
 n
product
nX
G[n]X
smash
GnX pn
Ganea space
Ganea – Whitehead framework for LS-category
cat(X)  n
 n admits a lifting
(Whitehead definition)
p n admits a section
(Ganea definition)
Many standard estimates of cat(X) can be derived from the above diagram. 12
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

13. TCn•(X) is the fibrewise category of the fibrewise pointed space
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
Iwase – Sakai: TC•(X) is the fibrewise LS category of the fibrewise pointed space
pr2 
X X×X X
Similarly:
TCn•(X) is the fibrewise category of the fibrewise pointed space
prn n
X Xn X
WnX
fibrewise fat wedge
nX X
 n
fibrewise product
nX
G[n]X
fibrewise smash product
GnX pn
fibrewise Ganea space
Ganea – Whitehead framework for monoidal topological complexity
Various explicit constructions: Iwase - Sakai, Fasso, Calcines, Doeraene, Vandembroucq, Franc - P. 13
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

14. Theorem (b) FIBREWISE POINTED OBSTRUCTION THEORY
FREE TOPOLOGICAL GROUPS AND FUNCTORIAL FIBREWISE CONSTRUCTIONS
(b) FIBREWISE POINTED OBSTRUCTION THEORY
Exploit functoriality of the Postnikov decomposition.
Let Pn: CRegB  TopB be the fibrewise pointed extension of the n-th Postnikov section functor Pn : CReg•  Top• B
For every X in CRegB the morphism X  PnX is an n-equivalence on the fibres. B
Theorem
If B is a finite dimensional complex and if Y is a fibrewise pointed space over B with fibres of dimension  l then
[Y,X]B  [Y, PnX]B
is a bijection for dimB+l < n and a surjection for dimB+l = n.
In other words X  PnX is a fibrewise pointed (n-dimB)-equivalence. 14
INTRODUCTION     CONSTRUCTION       APPLICATIONS   

15. THANK YOU! 15