at.algebraic topology - Properties of the class of topological spaces possessing a CW-structure

Let ${mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure).

Is it true that for a covering map $Estackrel{f}{to} B$ with $Ein{mathcal C}$ we have $Bin{mathcal C}$, too?

It is true that the total space of a covering lies in ${mathcal C}$ if the base space does, but the reverse implication is not clear to me.

Edit

As Algori pointed out, the quotient space is not even Hausdorff in general. What about finite regular coverings, i.e. those which come from a free action of a finite group on the total space? Is it true then that the quotient space carries a CW-structure, too?

I'm interested in that because this would imply that given a free group action of a finite group on a "nice" space like a CW-complex, one can always choose a CW-structure with respect to which $G$ just permutes cells. Then the corresponding cellular complex would be a (possibly nice) complex of ${mathbb Z}G$-modules (for example, if the space was a sphere, then this procedures can be used to construct a periodic ${mathbb Z}G$-resolution of the trivial module ${mathbb Z}$, showing that the group has to have periodic invariants like homology and cohomology; in this particular case, however, things behave well as the quotient space ${mathbb S}^n/G$ is still a compact manifold).

Thank you.

co.combinatorics - Generalizations of Planar Graphs

A well ordering, $leq$, on a set $S$ is a WELL-QUASI-ORDERING if and only if every sequence $x_iin S$ there exists some $i$ and $j$ natural numbers with $i < j$ with $x_ileq x_j$. (See wikipedia article at bottom)

Robertson-Seymour Theorem: The set $S=Graphs/isomorphism$ are well-quasi-ordered under contraction.

The corollary of this theorem is that any property $P$ of graphs which is closed under the relation of contraction (meaning if $P(G_2)$ and $G_1leq G_2$ then $P(G_1)$) is characterized by a finite set of excluded minors (Which is explained below). An example of such a $P$ is planarity, or linkless embeddability of a graph into R^3. i.e. Every contraction of a planar graph is planar.

Suppose $P$ is a property closed under $leq$.

***If $Bleq G$ and $B$ is not $P(B)$, Then not $P(G)$.

The idea is to characterize $P$ by a collection of bad $B$'s. The finiteness of the set of excluded minors comes from the well-quasi ordering and doesn't use the idea of a graph:
Assuming we have well-quasi-ordered set $(S,leq)$. One can prove that every property $P$ which is closed under the relation is characterized by a Finite set of excluded minors.That is, there exists some $X=lbrace x_1,ldots,x_nrbrace subset S $ such that for all $sin S$, $$ mbox{ not } P(s) iff exists i, x_i leq s $$.

The existence of a finite set $X$ is implicitly in 12.5 of Diestel (link at bottom, see the corollary of Graph Minor Theorem in Diestel). First convince yourself that there exists a set of $B's$ (not necessarily finite) as in * that characterize property $P$. Then consider the smallest such set of $B$'s and using the property of well-quasi ordering show that is is finite. Note that as in the second wikipedia article, we can say any set of elements $Asubset S$ such that for all $a,bin A$ we have $a nleq b$ must be a finite set (provided $leq$ is a well-quasi-ordering).

Actual work done showing stuff in a topological direction has be done by Eran Nevo http://www.math.cornell.edu/~eranevo/

I suspect that Matroids have a well-quasi-ordering and that there is work being done toward proving an analogous theorem for them.

I have a limit on links:

en.wikipedia.org/wiki/Well-quasi-ordering

en.wikipedia.org/wiki/Robertson-Seymour_theorem

diestel-graph-theory.com/GrTh.html

rt.representation theory - Finite dimensional spherical representation of $SO(n,1)(mathbb{R})$

Bonsoir Ludo! I am puzzled by the fact that your title asks something more restrictive than the OP, since the latter does not contain the word "spherical". Let me answer the latter first. Any finite-dimensional representation of $SO(n,1)(mathbb{R})$ extends to a representation of the complexification, which is $SO_{n+1}(mathbb{C})$. By Weyl's unitary trick, those are in 1-1 correspondence with unitary finite-dimensional representation of the maximal compact subgroup of the complexification, here $SO_{n+1}$. The finite-dimensional, unitary, irreducible representations of such a group are parametrized by their highest weight, and can be described via Verma modules, see Chapter IV in Knapp's "Representation theory of semi-simple groups" (Princeton UP, 1986).

Now, if you need only spherical irrep's, this amounts to consider irreducible $SO_{n+1}$-representations having non-zero $SO_n$-invariant vectors; or, equivalently (by an easy case of Frobenius reciprocity), irreducible $SO_{n+1}$-sub-representations of $L^2(S^n)$ (where $S^n=SO_{n+1}/SO_n$ is the $n$-sphere). These correspond to homogeneous harmonic polynomials in $n+1$ variables.

mg.metric geometry - How far can the analogy between a Cayley graph and a symmetric space be pushed?

The Cheeger constants for graphs and Riemannian locally symmetric spaces are closely related. Via inequalities of Buser and Cheeger, these are also related to eigenvalues of the laplacians for each. This analogy led to the first construction of expander graphs, by Margulis, via Property (T). More recently, this analogy has been exploited by several people, notably Marc Lackenby, to study finite-sheeted coverings using Cayley graphs of finite quotients as a finite simplicial approximation.

The point, roughly, is the following. Let $Gamma$ be a group with generating set $S$, and suppose $Gamma = pi_1(M)$ for some Riemannian manifold $M$. Then any finite quotient $F$ under a homomorphism $phi$ has a generating set $phi(S)$, so we can form the corresponding Cayley graph $mathcal{G}(F, phi(S))$. Properties of $mathcal{G}(F, phi(S))$ like girth, spectrum, expansion constants, Cheeger constant, and so forth are closely related to the analogous concept for the finite-sheeted covering $M_phi$ of $M$ corresponding to the subgroup $mathrm{kernel}(phi)$ of $Gamma$. This analogy is most potent when you consider a family {$mathcal{G}(F_j, phi_j(S))$} of Cayley graphs corresponding to a family $F_j$ of finite quotients of $Gamma$.

References for all these concepts are the books On Property ($tau$) by Lubotzky and Zuk (unpublished, but on Lubotzky's website), Discrete Groups, Expanding Graphs and Invariant Measures by Lubotzky, Elementary Number Theory, Group Theory and Ramanujan graphs by Davidoff, Sarnak, and Valette, and Marc Lackenby's paper Expanders, ranks and graphs of groups, Israel J. Math. 146 (2005) 357-370.

Fourier transform for dummies

One of the main uses of Fourier transforms is to diagonalize convolutions. In fact, many of the most useful properties of the Fourier transform can be summarized in the sentence "the Fourier transform is a unitary change of basis for functions (or distributions) that diagonalizes all convolution operators." I've been ambiguous about the domain of the functions and the inner product. The domain is an abelian group, and the inner product is the L² inner product with respect to Haar measure. (There are more general definitions of the Fourier transform, but I won't attempt to deal with those.)

I think a good way to motivate the definition of convolution (and thus eventually of the Fourier transform) starts with probability theory. Let's say we have an abelian group (G, +, -, 0) and two independent random variables X and Y that take values in G, and we are interested in the value of X + Y. For simplicity, let's assume G = {x₁, ..., x_n} is finite. For example, X and Y could be (possibly biased) six-sided dice, which we can roll to get two independent elements of Z/6Z. The sum of the die rolls mod 6 gives another element of the group.

For x ∈ G, let f(x) be the probability P(X = x), and let g(x) = P(Y = x). What we care about is h(x) := P(X + Y = x). We can compute this as a sum of joint probabilities:

h(x) = P(X + Y = x) = Σ_y+z=xP(X = y & Y = z)

However, since X and Y are independent, P(X = y & Y = z) = P(X = y)P(Y = z) = f(y)g(z), so the sum is actually

h(x) = Σ_y+z=xf(y)g(z) = Σ_y∈Gf(y)g(x-y).

This is called the convolution of f and g and denoted by f*g. In words, the convolution of two probability distributions is the probability distribution of the sum of two independent random variables having those respective distributions. From that, one can deduce easily that convolution satisfies nice properties: commutativity, associativity, and the existence of an identity. Moreover, convolution has the same relationship to addition and scalar multiplication as pointwise multiplication does (namely, bilinearity). In the finite setting, there's also an obvious L² inner product on distributions, with respect to which, for each f, the transformation g -> f * g is normal. Since such transformations also commute, recalling a big theorem from finite-dimensional linear algebra, we know there's an orthonormal basis with respect to which all of them are diagonal. It's not difficult to deduce then that in such a basis, convolution must be represented by coordinatewise multiplication. That basis is the Fourier basis, and the process of obtaining the coordinates in the Fourier basis from coordinates in the standard basis (the values f(x) for x ∈ G) is the Fourier transform. Since both bases are orthonormal, that transformation is unitary.

If G is infinite, then much of the above has to be modified, but a lot of it still works. (Most importantly, for now, the intuition works.) For example, if G = Rⁿ, then the sum Σ_y∈Gf(y)g(x-y) must be replaced by the integral ∫_y∈Gf(y)g(x-y)dy to define convolution, or even more generally, by Haar integration over G. The Fourier "basis" still has the important property of representing convolution by "coordinatewise" (or pointwise) multiplication and therefore of diagonalizing all convolution operators.

The fact that the Fourier transform diagonalizes convolutions has more implications than may appear at first. Sometimes, as above, the operation of convolution is itself of interest, but sometimes one of the arguments (say f) is fixed, and we want to study the transformation T(g) := f*g as a linear transformation of g. A lot of common operators fall into this category. For example:

• Translation: T(g)(x) = g(x-a) for some fixed a. This is convolution with a "unit mass" at a.


• Differentiation: T(g)(x) = g'(x). This is convolution with the derivative of a negative unit mass at 0.


• Indefinite integration (say on R): T(g)(x) = ∫^x_-infinityg(t)dt. This is convolution with the Heaviside step function.

In the Fourier basis, all of those are therefore represented by pointwise multiplication by an appropriate function (namely the Fourier transform of the respective convolution kernel). That makes Fourier analysis very useful, for example, in studying differential operators.

Tate Cohomology via Stable Categories

Situation

Let $G$ be a finite group and provide $Gtext{-mod} := {mathbb Z}Gtext{-mod}$ with the Frobenius structure of ${mathbb Z}$-split short exact sequences. Denote by $underline{Gtext{-mod}}$ the associated stable category with loop functor $Omega$.

For any Frobenius category $({mathcal A},{mathcal E})$ and a complete projective-injective resolution $P_{bullet}$ of some $Xin{mathcal A}$, we have for any $Yin{mathcal A}$ a canonical isomorphism of abelian groups

$H^n(text{Hom}_{mathcal A}(P_{bullet},Y))cong [Omega^n X,Y]$,

where $[-,-] := text{Hom}_{underline{{mathcal A}}}(-,-)$.

Applying this to $Gtext{-mod}$ yields an isomorphism

$widehat{H}^k(G;M)cong [Omega^k{mathbb Z},M]$,

where $widehat{H}^k(G;M)$ denotes the Tate-Cohomology of $G$ with values in $M$.

If I didn't mix things up, in this language Tate-Duality should mean that the canonical map

$[{mathbb Z},Omega^k{mathbb Z}]otimes_{mathbb Z}[Omega^k{mathbb Z},{mathbb Z}]to[{mathbb Z},{mathbb Z}]cong{mathbb Z}/|G|{mathbb Z}$

is a duality.

Question

I'd like to know sources which introduce and treat Tate cohomology in the way described above, i.e. using the language of Frobenius categories and its associated stable categories. In particular, I would be interested in a proof of Tate Duality using this more abstract language instead of resolutions.

Does anybody know such sources?

Remark

It seems to be more difficult to work over the integers instead of some field, for in this case, the exact sequences in the Frobenius structure $Gtext{-mod}$ are required to be ${mathbb Z}$-split, which is not automatic. As a consequence, there may be projective/injective objects in $(Gtext{-mod},{mathcal E}^{G}_{{e}})$ which are not projective/injective as ${mathbb Z}G$-modules. Further, the long exact cohomology sequence exists only for ${mathbb Z}$-split exact sequences of $G$-modules (not good, because Brown uses the exact sequence $0to {mathbb Z}to{mathbb Q}to{mathbb Q}/{mathbb Z}to 0$ in his proof of Tate duality); of course, one can choose particular complete resolutions of ${mathbb Z}$ consisting of ${mathbb Z}G$-projective modules, and such a resolution yields a long exact cohomology sequence for any short exact sequence of coefficient modules, but this seems somewhat unnatural and doesn't fit into the picture right now.

Partial Results

(1) For any subgroup $Hleq G$ there are restriction and corestriction morphisms

$[Omega^k {mathbb Z},-]^{underline{G}}=widehat{H}^*(G;-)leftrightarrowswidehat{H}^*(H;-)=[Omega^k{mathbb Z},-]^{underline{H}}$

defined as follows: for any $G$-module $M$, the abelian group $[{mathbb Z},M]^{underline{G}}$ is in canonical bijection with $M^G / |G| M^G$, and there are restriction and transfer maps

$text{res}: M^G / |G| M^Glongrightarrow M^H / |H| M^H,quad [m]mapsto [m]$,

$text{tr}: M^H / |H| M^Hlongrightarrow M^G / |G| M^Gquad [m]mapstoleft[sumlimits_{gin G/H} g.mright]$,

respectively. Now

$[Omega^k{mathbb Z},M]^{underline{G}}cong [{mathbb Z},Omega^{-k}M]^{underline{G}}stackrel{text{res}}{longrightarrow} [{mathbb Z},Omega^{-k}M]^{underline{H}}cong[Omega^k{mathbb Z},M]^{underline{H}}$

$[Omega^k{mathbb Z},M]^{underline{H}}cong [{mathbb Z},Omega^{-k}M]^{underline{H}}stackrel{text{tr}}{longrightarrow} [{mathbb Z},Omega^{-k}M]^{underline{G}}cong[Omega^k{mathbb Z},M]^{underline{G}}$

seems to be the natural thing to define restriction and transfer. (This is very similar to the usual method of giving a morphism of $delta$-functors only in degree $0$ and extend it by dimension shifting, though a bit more elegant in my opinion)

Note that it was implicitly used that $Omega^k$ commutes with the forgetful functor $Gtext{-mod}to Htext{-mod}$

(2) For any subgroup $Hleq H$, $gin G$ and a $G$-module $M$ there is a map

$g_*: widehat{H}^*(H;-)towidehat{H}^*(gHg^{-1};M)$

extending the canonical map

$M^H/|H|M^Hlongrightarrow M^{gHg^{-1}}/|H|M^{gHg^{-1}},quad [m]mapsto [g.m]$.

(1) and (2) fit together in the usual way; there is a transfer formula and a lifting criterion for elements of Sylow-subgroups.

(3) The cup product on $widehat{H}^*(G;{mathbb Z})$ is given simply by composition of maps:

$[Omega^p{mathbb Z},{mathbb Z}]otimes_{mathbb Z}[Omega^q{mathbb Z},{mathbb Z}]stackrel{Omega^qotimestext{id}}{longrightarrow}[Omega^{p+q}{mathbb Z},Omega^q{mathbb Z}]otimes_{mathbb Z}[Omega^q{mathbb Z},{mathbb Z}]longrightarrow [Omega^{p+q}{mathbb Z},{mathbb Z}]$

Does anybody see why this product is graded-commutative?

molecular biology - Is there a Reverse Transcription optimization for long, 9kb, transcripts?

Has anyone optimized RT for long transcripts (9kb)? The downstream application will be PCR amplification and Illumina library prep. It will be trivial to make internal primers sets for the PCR that are specific as long as there are no chimeric sequences. If there are, they will probably get primed also. If anyone knows of an optimization and/or other potential pitfalls, I would love to hear them.