Download CONCERNING INVERSIBLE FIBRATIONS, GLOBALIZATION AND

CONCERNING INVERSIBLE FIBRATIONS,
GLOBALIZATION AND TRIVIALITY OF MAPS
NIGAR TUNCER OZARSLAN
SAMUEL EILENBERG CENTENARY CONFERENCE,2013
WARSAW, POLAND
SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND
NIGAR T. OZARSLAN
Question Let
be a locally trivial mapping and
suppose that B is contractible. Does it follow that p is
trivial?
Definition The mapping
is called trivial if there
exists a homeomorphism
for some space
such that
Corollary Assume that the space is paracompact,
Hausdorff and locally contractible in the large. Then
is an inversible fibration if and only if is locally trivial.
E. Dyer and S.Eilenberg, Globalizing fibrations with
schedules, Fund. Math. 130 (1988)
SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND
NIGAR T. OZARSLAN
Definition Let
be a map and
be a
space over and
.
Let
such that
. is called
Hurewicz fibration if and only if there exists a map
such that
is the identity map
Definition Let denote the set of all real numbers
The path space
of a topological space is the
subspace of
defined by
Let
be a space over . The space
is the
subspace of
of all pairs
such that
A mapping is a D&E fibration if and only if there exists
a continuous function (called action)
To be written
which satisfies the conditions
and
, where
SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND
NIGAR T. OZARSLAN
A mapping is called inversible fibration if and only if there
exists two reciprocal actions
satisfying
for all
defined as
Where the inverse path is
SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND
NIGAR T. OZARSLAN
Regularity, inversibility, D&E fibrations, Hurewicz fibration
Definition A lifting function is called regular if it lifts
constant paths to constant paths.
Definition An action is called regular action if and only if
it satisfies the condition
for all
where
Definition A map is an inversible Hurewicz fibration if
and only if there exists two lifting functions
satisfying
.
for every
Definition A space over B is called regularly inversible
Hurewicz fibration if and only if it is an inversible
Hurewicz fibration with reciprocal regular lifting
functions.
SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND
NIGAR T. OZARSLAN
Theorem A space over B is an inversible D&E fibration
then it is inversible Hurewicz fibration.
Theorem A space over B is a regular D&E fibration if and
only if it is a regular Hurewicz fibration.
Theorem A space is regularly inversible D&E fibration if
and only if it is regularly inversible fibration.
Theorem If B is paracompact and contractible and f is a
space over B then the followings are equivalent;
(1) F is locally trivial,
(2) F is an inversible D&E fibration,
(3) F is an inversible Hurewicz fibration,
(4) f is trivial.
SAMUEL EILENBERG CENTENARY CONFERENCE, 2013, WARSAW, POLAND
THANK YOU
NIGAR T. OZARSLAN