4 Feb 2011 20:53

The starting point is that the unitary group U is connected but not simply connected: its \pi_1 is Z. Then its universal covering is a principal Z-bundle, so U is naturally equipped with a map to BZ; delooping this map we get a characteristic map BU --> B^2Z. This is the first Chern characteristic map, and its existence is topologically neat and obviuos.

The topological realization of c_2, on the other hand, should be a natural map BU --> B^4Z, and this is less obvious. but by Bott periodicity, U \simeq \Omega2 U, so in particular the double loop space Omega^2U is connected and simply connected, and so it is equipped with a characteristic map to BZ. The double loop group \Omega^2U is three times deloopable, so we get a characteristic map BU --> B^4Z.

question: is this the second Chern class?

it's at least a c_2 + b (c_1)^2
but how to determine those coefficients?
reference?

jim

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5 Feb 2011 15:51

A graded Hall basis is implicit in Hilton's On the homotopy groups of
the union of spheres

Does anyone have a reference to an abstract defn of graded Hall basis

jim

5 Feb 2011 19:03

dear jim:  i have no idea what is meant by an "abstract definition" of a graded hall basis.  but i think i know how to produce one in two ways, depending on what you mean.  both ways involve some choices.

1)  you could mean a basis for the graded free lie algebra LV on a graded vector space V.  that is probably what you mean but it is not what hilton meant and used!  he used the classical case.  in some sense, he wanted a geometric basis for the wedge X \vee Y.  you get this in the classical ungraded way.  see 2 below.

i shall assume that 2 is a unit in the ground ring .  it is simpler then.  the reference for all this is my book which also tells you how to proceed when 2 is not a unit.

if you mean the true graded case above, some preliminaries should be recalled where the graded case is different from the ungraded case.

in the graded case, recall that,

a)  if x has odd degree, Lx = <x,[x,x]} has a basis of two elements and ,
b)  if x has even degree, Lx = <x> has a basis of one element.

in either case, we start by picking off one element x as a basis element and see what we have left.  we do this as follows.

consider the exact sequence
0 \to K \to LV \to <x> \to 0 where
I) x is one of the basis elements of V, write V = <x> \oplus W and (it can be shown that)
II) K is a free graded lie algebra with generating graded vector space as follows

a)  if x has even degree, the generating vector space for K is
Z = W \oplus [x,W] \oplus [x,[x,W]] \oplus ......, that is, K = LZ where Z is the direct sum

b)  if x has odd degree, the generating vector space for K is
Z = <[x,x]> \oplus W \oplus [x,W], that is, K = LZ

note:  you can see this in the trivial one dimensional case LV = Lx where W is 0.

now iterate this, picking off a generator of Z in the same way.
the result is the (usually infinite) hall basis for LV

for example, if LV = L<x,y> is generated by two elements with x even and y odd, you get
x, y, [x,y],[x[x,y]], ....
then [y,y], [y,[x,y]],[y,[x,[x,y]]], ...

continue picking off one at a time to grow the list.
in the classical ungraded case (that is, everything is concentrated in even degree) hall gave a more specific method of choice for the order of picking off one generator at a time.

2)  but you might not mean the graded case.  you might mean the usual ungraded hall basis. after all this is what hilton used and milnor also.  the hilton-milnor theorem is based on the following variation.  suppose L = LV where V is a graded vector space.  write V = W \oplus U as a direct sum.

consider the exact sequence
0 \to K \to L \to LW \to 0
where K is the free graded lie algebra with generating basis Z =
U \oplus [W,U] \oplus [W,[W,U]] \oplus ..... = \bigoplus_{k \geq 0} ad^k (U) (W), that is, K = LZ

hilton's choice of a hall basis is the result of iterating this, picking off one summand at a time, starting with U, then [W,U], then [W,[W.U]], etc
if U is nonzero, this is produces a countable collection of vector spaces.
note that this is really just the ungraded case of 1, that is, where everything is concentrated in even degrees.  it is therefore the classical case.

closing remarks:  if 2 is not a unit in the ground ring and x has odd degree, then [x,x] has to be replaced by the square x^2.  (note that [x,x] = xx + xx = 2x^2 when x has odd degree.)

as i said before, the reference is my new book, "algebraic methods in unstable homotopy theory", cambridge university press.

best wishes,

joe neisendorfer

On Sat, Feb 5, 2011 at 9:51 AM, jim stasheff wrote:
A graded Hall basis is implicit in Hilton's On the homotopy groups of the union of spheres

Does anyone have a reference to an abstract defn of graded Hall basis for a graded Lie algebra?

jim

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5 Feb 2011 19:13

dear jim:  ooops!  in my previous message, you really start with W, then U, [W,U], [W, [W,U]],  etc.

i forgot to mention the starting W.  sorry.
joe neisendorfer

On Sat, Feb 5, 2011 at 1:03 PM, joseph neisendorfer wrote:
dear jim:  i have no idea what is meant by an "abstract definition" of a graded hall basis.  but i think i know how to produce one in two ways, depending on what you mean.  both ways involve some choices.

1)  you could mean a basis for the graded free lie algebra LV on a graded vector space V.  that is probably what you mean but it is not what hilton meant and used!  he used the classical case.  in some sense, he wanted a geometric basis for the wedge X \vee Y.  you get this in the classical ungraded way.  see 2 below.

i shall assume that 2 is a unit in the ground ring .  it is simpler then.  the reference for all this is my book which also tells you how to proceed when 2 is not a unit.

if you mean the true graded case above, some preliminaries should be recalled where the graded case is different from the ungraded case.

in the graded case, recall that,

a)  if x has odd degree, Lx = <x,[x,x]} has a basis of two elements and ,
b)  if x has even degree, Lx = <x> has a basis of one element.

in either case, we start by picking off one element x as a basis element and see what we have left.  we do this as follows.

consider the exact sequence
0 \to K \to LV \to <x> \to 0 where
I) x is one of the basis elements of V, write V = <x> \oplus W and (it can be shown that)
II) K is a free graded lie algebra with generating graded vector space as follows

a)  if x has even degree, the generating vector space for K is
Z = W \oplus [x,W] \oplus [x,[x,W]] \oplus ......, that is, K = LZ where Z is the direct sum

b)  if x has odd degree, the generating vector space for K is
Z = <[x,x]> \oplus W \oplus [x,W], that is, K = LZ

note:  you can see this in the trivial one dimensional case LV = Lx where W is 0.

now iterate this, picking off a generator of Z in the same way.
the result is the (usually infinite) hall basis for LV

for example, if LV = L<x,y> is generated by two elements with x even and y odd, you get
x, y, [x,y],[x[x,y]], ....
then [y,y], [y,[x,y]],[y,[x,[x,y]]], ...

continue picking off one at a time to grow the list.
in the classical ungraded case (that is, everything is concentrated in even degree) hall gave a more specific method of choice for the order of picking off one generator at a time.

2)  but you might not mean the graded case.  you might mean the usual ungraded hall basis. after all this is what hilton used and milnor also.  the hilton-milnor theorem is based on the following variation.  suppose L = LV where V is a graded vector space.  write V = W \oplus U as a direct sum.

consider the exact sequence
0 \to K \to L \to LW \to 0
where K is the free graded lie algebra with generating basis Z =
U \oplus [W,U] \oplus [W,[W,U]] \oplus ..... = \bigoplus_{k \geq 0} ad^k (U) (W), that is, K = LZ

hilton's choice of a hall basis is the result of iterating this, picking off one summand at a time, starting with U, then [W,U], then [W,[W.U]], etc
if U is nonzero, this is produces a countable collection of vector spaces.
note that this is really just the ungraded case of 1, that is, where everything is concentrated in even degrees.  it is therefore the classical case.

closing remarks:  if 2 is not a unit in the ground ring and x has odd degree, then [x,x] has to be replaced by the square x^2.  (note that [x,x] = xx + xx = 2x^2 when x has odd degree.)

as i said before, the reference is my new book, "algebraic methods in unstable homotopy theory", cambridge university press.

best wishes,

joe neisendorfer

On Sat, Feb 5, 2011 at 9:51 AM, jim stasheff wrote:
A graded Hall basis is implicit in Hilton's On the homotopy groups of the union of spheres

Does anyone have a reference to an abstract defn of graded Hall basis for a graded Lie algebra?

jim

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6 Feb 2011 18:42

### Fwd: Computational and Applied Topology in Utah


There will be a one week program in Computational and Applied Topology
as part of the Mathematics Research Communities program sponsored by the
AMS. It will take place during the period June 19-25, 2011 at Snowbird, Utah.
The idea is to introduce new Ph.D.'s and graduate students to this
emerging area of mathematics.   It is open to young mathematicians,
including graduate students within 2 years of receiving the Ph.D. and
new Ph.D.'s within 3 years of their doctorate.  It will be organized by
Rob Ghrist, Ben Mann, and Gunnar Carlsson.  Please see the link at

http://www.ams.org/programs/research-communities/mrc-11

Gunnar Carlsson

6 Feb 2011 20:33

### Job in Münster

Dear all,

the WWU Münster is advertising a permanent professorship with a strong
preference for topology. See

http://www.uni-muenster.de/Rektorat/Stellen/st_3924.htm

A new collaborative research center has recently been awarded to the
math department in Münster. Because of this it is relatively easy to
obtain 3rd party funding for Phd students and post-docs through the DFG.

There is no formal language requirement, in particular initially
teaching could be in English.  This job is at the W2 level which means a
starting salary of at least Euro  4.193,25 per month.  The deadline for
applications is march 15.

Feel free to contact me with questions regarding this job:
a.bartels@...

-Arthur Bartels

7 Feb 2011 15:58

### Conference on "toric methods" in Belfast, July 2011


Registration has now started for the conference

"Toric methods in homotopy theory and related subjects"

to take place in in Queen's University Belfast (Northern Ireland, UK),
18-20 July 2011. Confirmed speakers include Jack Morava, Taras Panov, Sam
Payne, Oliver Roendigs, and Alex Suciu.

General information and registration instruction can be found at the

http://toricmethodsbelfast. zzl. org/

The meeting is supported by EPSRC grant EP/H018743/1, and an LMS
conference grant.

Please forward this message to anyone who might be interested!

Thomas Huettemann and Nigel Ray

Contact: toric@...

7 Feb 2011 19:04

### arbeitsgemeinschaft

Everyone

This is the second (and final) announcement of the next Arbeitsgemeinschaft at
Oberwolfach. Please pass the word to anyone who might be interested. Thanks.

Arbeitsgemeinschaft mit aktuellem Thema:

Rational Homotopy in Mathematics and Physics

at the Mathematisches Forschungsinstitut Oberwolfach

The next Arbeitsgemeinschaft will be held at the Mathematisches
Forschungsinstitut Oberwolfach from April 3, 2011 to April 8, 2011.

their descriptions and references, go to the MFO website
www.mfo.de
and click on Arbeitsgemeinschaft. You can also get there through

There will be two survey talks given by experts in
rational homotopy and in geometry. Talk 1, Fundamentals of Geometry,
will be given by Wilderich Tuschmann and Talk 18, Differential Modules
and Applications, will be given by Yves Felix. All other talks need
volunteers. The titles of the talks in the program
are:

1. Fundamentals of Geometry
2. Sullivan models
3. Group Actions
4. Geodesics and the Free Loop Space I
5. Geodesics and the Free Loop Space II
6. Geodesic Flows
7. Formality and Kahler Manifolds
8. Spectral Sequences and Models
9. Formality and Symplectic Manifolds I
10. Formality and Symplectic Manifolds II
11. Curvature I
12. Curvature II
13. Poincare Duality and Models I.
14. Poincare Duality and Models II
15. String Topology I
16. String Topology II
17. Chen's Iterated Integrals and higher Hochschild chain complex
18. Differential Modules and Applications

If you would like to participate, please send your full name and full
postal address to both John Oprea and Daniel Tanre at

j.oprea@...   and    Daniel.Tanre@...

by  February 13, 2011  at the latest.

You should also indicate which talk you are willing to give:

First choice: talk no. .....

Second choice: talk no. .....

Third choice: talk no. .....

is possible and whether you have been chosen to give one of the lectures.

The Arbeitsgemeinschaft will take place at
Mathematisches Forschungsinstitut Oberwolfach,
Lorenzenhof, 77709 Oberwolfach-Walke, Germany.
The institute offers accommodation free of charge to the participants.
Travel expenses cannot be covered. Further information will be given to

John Oprea
Department of Mathematics
Cleveland State University

Is not the most beautiful mathematics "comme la rencontre fortuite sur
une table de dissection d´une machine à coudre et d´un parapluie !"?

15 Feb 2011 19:07

### Announcement: Young Topologists Meeting, Lausanne Switzerland June 14-18, 2011

Dear all,

This is the second announcement of the next Young Topologists Meeting.

We are pleased to announce that the 2011 Young Topologists Meeting
will take place June 14-18, 2011 at the EPFL in Lausanne, Switzerland.

The goal of the YTM is to bring together graduate students, recent
doctoral recipients and other young researchers in topology to give
talks on their own work and meet others in the field.

In addition, we will be presenting a lecture series by Birgit Richter (University of
Hamburg) entitled "Algebraic properties of ring spectra". Her lectures will describe aspects of ring spectra that concern their algebraic features such as Picard groups, Galois theory (a la Rognes), Brauer groups and algebraic K-theory.

We expect to have funds available to support some participants.

More information and a registration form can be found on the YTM
website at http://sma.epfl.ch/~finster/ytm/ytm.html

We hope to see you this summer in Switzerland!

Eric Finster
Varvara Karpova
Marc Stephan

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15 Feb 2011 20:18

### (no subject)

I have a question for the list.

What is known on analogs of Lefschetz hyperplane theorem for Kahler manifolds.

I know that there are analogs of symplectic manifolds (Donaldson) and something on Kahler manifolds with non-negative curvature. What else?

Dr. Yuli B. Rudyak Department of Mathematics University of Florida 358 Little Hall PO Box 118105 Gainesville, FL 32611-8105 USA TEL: (+1) 352-392-0281 ext. 319(office) TEL: (+1) 352-381-8497(home) FAX: (+1) 352-392-8357 URL: http://www.math.ufl.edu/~rudyak/
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Gmane