Subject: Re: two postings
From: "John R. Klein"
Date: Mon, 9 Jul 2007 09:27:45 -0400
Hi Jim,
That's interesting. I suspect the reason for the second choice
has to do with Dirac's bracket notation f(x) = , in which
mathematicians
usually prefer to write the covector f on the left (because it is in
accord with
the notation for evaluating a function on an element).
j.
> Subject: item for the discussion list
> From: James E McClure
> Date: Sat, 7 Jul 2007 10:39:29 -0400 (EDT)
>
> I spent some time recently untangling a confusion about Poincare duality
> and I
> thought it might be helpful for other people to know about it.
>
> There are two ways to define the Poincare duality map from S^*M to S_*M:
> either take x to o\cap x or to x \cap o (where o is a cycle representing
the
> fundamental class). All the textbooks I've looked at (except for
Ranicki's
> book Algebraic and Geometric Surgery) use the second choice, but it
turns
> out that the first choice is better. For example, if we convert S^*M to
a
> chain complex T_*M in the usual way (T_pM=S^{-p}M) then the first choice
for
> Poincare duality gives a chain map (raising degrees by dim M) from T_*M
to
> S_*M but the second does not (it {\it commutes} with the differential,
which
> is not what one wants for a chain map raising degrees by dim M). Dold's
> book
> Algebraic Topology uses the wrong choice, and Dold then observes that
> there is
> an anomaly in the definition of the umkehr map (page 314 and the top of
page
> 315); this anomaly would not have occurred if he had made the right
> choice for
> Poincare duality.
>
John R. Klein, Professor
Department of Mathematics
Wayne State University
Room 1213 FAB, 656 W. Kirby
voice: (313) 577-3174
fax: (313) 577-7596
_____________________________________________________________________
Subject: Slant products
From: Peter May
Date: Tue, 10 Jul 2007 11:14:24 -0500
This is a response to Jim McClure's posting. As I read
it, he presupposes a commonly accepted definition of slant
products. It is not at all clear to me that there is one.
The older literature is a quagmire, and I explicitly warn
against Spanier's book, where the two slant products (slash
and backslash, / and \bs below) are defined as if they differ
only by a sign. I include below the tex file of the last
section of my paper
The additivity of traces in triangulated categories.
Advances in Mathematics 163(2001), 34--73.
The pdf file of that paper is posted on my web page:
http://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf
The point of the extract that follows is that slant products
can best be systematized in the general framework of (closed)
symmetric monoidal categories with a compatible triangulation,
such as the stable homotopy category and classical derived
categories. For the compatibility with Poincare duality, the
latter is best viewed as Spanier-Whitehead duality plus the
Thom isomorpism. Chain level adaptations should be clear.
\section{Homology and cohomology theories}
When $\sC$ is the stable homotopy category, one can give a
general treatment of the products in homology and cohomology
theories that is based solely on the structure of $\sC$ as a
symmetric monoidal category with a compatible triangulation.
There are four basic products here, two of which are called
``slant products''. A systematic exposition is given by Adams
\cite[III\S9]{Adams} [Stable homotopy and generalized homology]
and followed by Switzer \cite[pp. 270--284]{Switzer}. We warn
the reader that the treatment of slant products in the literature
is chaotic. No other two sources seem to give the same signs, and
some standard references actually confuse the slant product $\bs$
with a product that differs only by a sign from the slant product
$/$. We run through a version of Adams' definitions and pinpoint
the role played by the new axioms. If we were starting from scratch,
our preferred version of slant products would differ by signs from
those below, but the logical advantage of writing variables in their
most natural order is outweighed by the need for consistency in the
literature. Adams and Switzer make no use of function spectra $F(X,Y)$,
which were only obtainable by use of Brown's representability theorem
at the time they were writing, and this obscures the formal nature of
their definitions of the products.
For an object $X$ of $\sC$ and an integer $n$, define
$$\pi_n(X) = \sC(S^n,X).$$
When $\sC$ is the stable homotopy category, $\pi_n(X)$ is the
$n$th homotopy group of the spectrum $X$. When $\sC$ is the
derived category of chain complexes over a commutative ring $R$,
$S^n$ is the trivial chain complex given by $R$ in degree $n$ and
$\pi_n(X)$ is the $n$th homology group of the chain complex $X$.
Applying the product $\sma$ ($\ten$ in algebraic settings), we
obtain a natural pairing
\begin{equation}\label{pair}
\pi_m(X)\ten \pi_n(Y) \rtarr \pi_{m+n}(X\sma Y).
\end{equation}
For objects $X$ and $E$, algebraic topologists define
$$ E_n(X) = \pi_n(E\sma X) \ \ \text{and} \ \ E^n(X) = \pi_{-n}F(X,E).$$
Equivalently, $E^n(X) \iso \sC(X,\SI^n E)$. The four products
referred to above are
\begin{equation}\label{1}
\sma: D_p(X)\ten E_q(Y) \rtarr (D\sma E)_{p+q}(X\sma Y),
\end{equation}
\begin{equation}\label{2}
\cup: D^p(X)\ten E^q(Y) \rtarr (D\sma E)^{p+q}(X\sma Y),
\end{equation}
\begin{equation}\label{3}
/: D^p(X\sma Y)\ten E_q(Y) \rtarr (D\sma E)^{p-q}(X),
\end{equation}
\begin{equation}\label{4}
\bs: D^p(X)\ten E_q(X\sma Y) \rtarr (D\sma E)_{q-p}(Y).
\end{equation}
The naturality of slant products is better seen by rewriting
them in adjoint form
\begin{equation}\label{3'}
/: D^p(X\sma Y)\rtarr \Hom(E_q(X),(D\sma E)^{p-q}(X)),
\end{equation}
\begin{equation}\label{4'}
\bs: E_q(X\sma Y)\rtarr \Hom(D^p(X),(D\sma E)_{q-p}(Y)).
\end{equation}
The four products are obtained by passing to $\pi_*$ and applying
the pairing (\ref{pair}) and functoriality, starting from formally
defined canonical maps
\begin{equation}\label{1a}
D\sma X\sma E\sma Y \rtarr D\sma E\sma X\sma Y,
\end{equation}
\begin{equation}\label{2a}
F(X,D)\sma F(Y,E)\rtarr F(X\sma Y,D\sma E),
\end{equation}
\begin{equation}\label{3a}
F(X\sma Y,D)\sma E\sma Y \rtarr F(X, D\sma E),
\end{equation}
\begin{equation}\label{4a}
F(X,D)\sma E\sma X\sma Y \rtarr D\sma E\sma Y.
\end{equation}
Here (\ref{3a}) is obtained by permuting $E$ and $Y$ and using the
natural isomorphism
$$F(X\sma Y,D)\iso F(Y,F(X,D)),$$
the evaluation map $\epz: F(Y,F(X,D))\sma Y\rtarr F(X,D)$, and the
natural map
$$\nu: F(X,D)\sma E\rtarr F(X,D\sma E),$$
while (\ref{4a}) is obtained by permuting $E$ and $X$ and using the
evaluation map
$\epz: F(X,D)\sma X\rtarr D$.
Of course, when $D=E$ is a monoid in $\sC$ (ring spectrum in the
algebraic topology setting), we can compose the given external
products with maps induced by the product $E\sma E\rtarr E$ to
obtain internal products. Similarly, when $X=Y$ has a coproduct
$X\rtarr X\sma X$ or product $X\sma X\rtarr X$, we can obtain
internal products by composition. In topology, we are thinking of
reduced cohomology and the diagonal map
$A_+ \rtarr (A\times A)_+\iso A_+\sma A_+$ on spaces $A$.
The internalization of the product $\bs$ is the cap product.
There are many unit, associativity, and commutativity relations
relating the four products, and these are catalogued in \cite{Adams}
and \cite{Switzer}. Without exception, these formulas are direct
consequences of our axioms for a symmetric monoidal category with a
compatible triangulation. In particular, Adams \cite[pp. 235--244]{Adams}
and Switzer \cite[pp. 276--283]{Switzer} catalogue many formulas and
commutative diagrams that relate the four products to the connecting
homomorphisms in the homology and cohomology of pairs $(X,A)$ and $(Y,B)$,
the crucial point being the correct handling of signs. Modulo change of
notation, they are considering the behavior of smash products and
function spectra with respect to pairs of distinguished triangles in the
stable homotopy category. Our compatibility axioms give what is needed
to make the derivations of these formulas and diagrams formal consequences
of the axioms.