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 $$\label{pair} \pi_m(X)\ten \pi_n(Y) \rtarr \pi_{m+n}(X\sma Y).$$ 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 $$\label{1} \sma: D_p(X)\ten E_q(Y) \rtarr (D\sma E)_{p+q}(X\sma Y),$$ $$\label{2} \cup: D^p(X)\ten E^q(Y) \rtarr (D\sma E)^{p+q}(X\sma Y),$$ $$\label{3} /: D^p(X\sma Y)\ten E_q(Y) \rtarr (D\sma E)^{p-q}(X),$$ $$\label{4} \bs: D^p(X)\ten E_q(X\sma Y) \rtarr (D\sma E)_{q-p}(Y).$$ The naturality of slant products is better seen by rewriting them in adjoint form $$\label{3'} /: D^p(X\sma Y)\rtarr \Hom(E_q(X),(D\sma E)^{p-q}(X)),$$ $$\label{4'} \bs: E_q(X\sma Y)\rtarr \Hom(D^p(X),(D\sma E)_{q-p}(Y)).$$ The four products are obtained by passing to $\pi_*$ and applying the pairing (\ref{pair}) and functoriality, starting from formally defined canonical maps $$\label{1a} D\sma X\sma E\sma Y \rtarr D\sma E\sma X\sma Y,$$ $$\label{2a} F(X,D)\sma F(Y,E)\rtarr F(X\sma Y,D\sma E),$$ $$\label{3a} F(X\sma Y,D)\sma E\sma Y \rtarr F(X, D\sma E),$$ $$\label{4a} F(X,D)\sma E\sma X\sma Y \rtarr D\sma E\sma Y.$$ 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.