Obstructions to Compatible
Extensions of Mappings
Jose Perea
Duke University
Joint with John Harer
June 1994
Monday
(05/26/2014)
June 1994
Incremental
‘s
Monday
(05/26/2014)
June 1994
Incremental
‘s
Monday
(05/26/2014)
June 1994
Incremental
‘s
2002
Topological
Persistence
Monday
(05/26/2014)
June 1994
Incremental
‘s
2002
Topological
Persistence
2005
Computing
P.H.
Monday
(05/26/2014)
June 1994
Incremental
‘s
2002
Topological
Persistence
2005
Computing
P.H.
2008
Extended
Persistence
Monday
(05/26/2014)
…
June 1994
Incremental
‘s
2002
Topological
Persistence
2005
Computing
P.H.
2008
2009
Extended
Zig-Zag
Persistence Persistence
Monday
(05/26/2014)
June 1994
Incremental
‘s
2002
Topological
Persistence
2005
Computing
P.H.
2008
2009
Extended
Zig-Zag
Persistence Persistence
Monday
(05/26/2014)
Study the whole
multi-scale object at once
Is not directionality, but
compatible choices
…
…
What have we learned?
For Point-cloud data:
1. Encode multi-scale information in a filtration-like object
2. Make compatible choices across scales
3. Rank significance of such choices
The Goal:
To leverage the power of
the relative-lifting paradigm
and the language of obstruction theory
The Goal:
To leverage the power of
the relative-lifting paradigm
and the language of obstruction theory
For data analysis!
Why do we care?
Useful concepts/invariants can be interpreted this way:
1. The retraction problem:
2. Extending sections:
3. Characteristic classes.
Back to Point-clouds:
Model fitting
Example (model fitting):
(Klein bottle
model)
(3-circle model)
Mumford Data
Model fitting
Only birth-like events
Local to global
Example: Compatible
extensions of sections
Local to global
Only death-like events
Model fitting
Local to global
Combine the two:
The compatible-extension problem
How do we set it up?
Definition : The diagram
Extends compatibly, if there exist extensions
of the
so that
.
For instance :
Let
If
be the tangent bundle over
then
, and fix classifying maps
, where
Thus,
Extend separately
but
not compatibly
Let
If
be the tangent bundle over
then
, and fix classifying maps
, where
Thus,
Extend separately
but
not compatibly
Let
If
be the tangent bundle over
then
, and fix classifying maps
, where
Thus,
Extend separately
but
not compatibly
Let
If
be the tangent bundle over
then
, and fix classifying maps
, where
Thus,
Extend separately
but
not compatibly
Observation:
Compatible extension problem
Relative lifting problem
up to homotopy rel
How do we solve it?
Solving the classic extension problem:
The set-up
Assume
Want
Solving the classic extension problem:
The set-up
Assume
Want
Solving the classic extension problem:
The set-up
Assume
Want
Solving the classic extension problem:
The obstruction cocycle
Assume
Want
Theorem
is a cocycle, and
if and only if
extends. Moreover, if
for some
then there exists a map
so that
on
, and
Theorem
is a cocycle, and
if and only if
extends. Moreover, if
for some
then there exists a map
so that
on
, and
Solving the compatible extension problem:
The set-up
Assume
Theorem I (Perea, Harer)
Let
Then
which is zero if and only if
for some
.
is a cocycle,
Theorem II (Perea, Harer)
Let
. If
for
and
, then
extend compatibly.
The upshot:
Once we fix
so that
then
parametrizes the redefinitions of
extend. Moreover, if a pair
satisfies
via
that
,
then the redefinitions of
and
,
, extend compatibly.
and
The upshot:
Once we fix
so that
then
parametrizes the redefinitions of
extend. Moreover, if a pair
satisfies
via
that
,
then the redefinitions of
and
,
, extend compatibly.
and
Putting everything together
…
Example
Can we actually
compute this thing?
Can we actually
compute this thing?
Can we actually
compute this thing?
* Some times
Coming soon:
• Applications to database consistency
• Topological model fitting
• Bargaining/consensus in social networks
Thanks!!