diff --git a/thesis/notes/Basics.tex b/thesis/notes/Basics.tex index 192c6ce..0c03b2a 100644 --- a/thesis/notes/Basics.tex +++ b/thesis/notes/Basics.tex @@ -67,20 +67,24 @@ The following two theorems can be found in textbooks about homological algebra s \Theorem{kunneth}{ (Künneth Theorem) For spaces $X$ and $Y$, there is a short exact sequence - $$ 0 \to H(X; A) \tensor H(Y; A) \to H(X \times Y; A) \to \Tor(H(X; A), H(Y; A)) \to 0, $$ - where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded. \todo{Geef algebraische versie voor ketencomplexen} + \[ \footnotesize \xymatrix @C=0.3cm{ + 0 \ar[r] & H(X; A) \tensor H(Y; A) \ar[r] & H(X \times Y; A) \ar[r] & \Tor_{\ast-1}(H(X; A), H(Y; A)) \ar[r] & 0 + },\] + where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded. \todo{Geef algebraische versie voor ketencomplexen? en cohomology?} } -\section{Immediate results for rational homotopy theory} +\section{Consequences for rational homotopy theory} The latter two theorems have a direct consequence for rational homotopy theory. By taking $A = \Q$ we see that the torsion groups vanish. We have the immediate corollary. \Corollary{rational-corollaries}{ - We have the following natural isomorphisms in homology - $$ H(X) \tensor \Q \tot{\iso} H(X; \Q), $$ - $$ H(X \times Y; \Q) \tot{\iso} H(X; \Q) \tensor H(Y; \Q). $$ - Furthermore we can relate homology and cohomology in a natural way: - $$ H^n(X; \Q) \tot{\iso} \Hom(H_n(X); \Q). $$ + We have the following natural isomorphisms in rational homology, and we can relate rational cohomolgy naturally to rational homology + \begin{align*} + H_\ast(X) \tensor \Q &\tot{\iso} H_\ast(X; \Q), \\ + H_\ast(X; \Q) \tensor H_\ast(Y; \Q) &\tot{\iso} H_\ast(X \times Y; \Q), \\ + H^\ast(X; \Q) &\tot{\iso} \Hom(H_\ast(X); \Q). + \end{align*} + } The long exact sequence for a Serre fibration also has a direct consequence for rational homotopy theory. diff --git a/thesis/notes/Serre.tex b/thesis/notes/Serre.tex index 0a8e3d9..30c2655 100644 --- a/thesis/notes/Serre.tex +++ b/thesis/notes/Serre.tex @@ -35,10 +35,10 @@ Note that the maps $0 \to C$ and $C \to 0$ are $\C$-isomorphisms for any $C \in In the following arguments we will consider fibrations and need to compute homology thereof. Unfortunately there is no long exact sequence for homology of a fibration, however the following lemma expresses something similar. It is usually proven with spectral sequences, \cite[Ch. 2 Thm 1]{serre}. However in \cite{kreck} we find a more geometric proof. \Lemma{kreck}{ - Let $\C$ be a Serre class. Let $p: E \fib B$ be a fibration between $0$-connected spaces and $F$ its fiber. If $\RH_i(F) \in \C$ for all $i < n$, then + Let $\C$ be a Serre class and $p: E \fib B$ be a fibration between $0$-connected spaces with a $0$-connected fiber $F$. If $\RH_i(F) \in \C$ for all $i < n$ and $B$ is $m$-connected, then \begin{itemize} - \item $H_i(E, F) \to H_i(B, b_0)$ is a $\C$-iso for $i \leq n+1$ and - \item $H_i(E) \to H_i(B)$ is a $\C$-iso for all $i \leq n$. + \item $H_i(E, F) \to H_i(B, b_0)$ is a $\C$-iso for $i \leq n+m$ and + \item $H_i(E) \to H_i(B)$ is a $\C$-iso for all $i < n+m$. \end{itemize} } \Proof{ @@ -61,19 +61,20 @@ In the following arguments we will consider fibrations and need to compute homol As we are working with relative homology $H_{i+1}(E^{k+1}, E^k)$, we only have to consider the interiors of the $k+1$-cells (by excision). Each interior of a $k+1$-cell is a product, as $p$ was a fiber bundle. So we note that we have an isomorphism: $$ H_{i+1}(E^{k+1}, E^k) \iso H_{i+1}(\coprod_\alpha D^{k+1}_\alpha \times F, \coprod_\alpha S^k_\alpha \times F). $$ - Now we can apply the Künneth theorem and identify the cells of $B$: - \begin{align*} - H_{i+1}(E^{k+1}, E^k) &\iso (H_\ast(\coprod_\alpha D^{k+1}_\alpha, \coprod_\alpha S^k_\alpha) \tensor H_\ast(F))_{i+1} \\ - &\iso (H_\ast(B^{k+1}, B^k) \tensor H_\ast(F)) \\ - &= \bigoplus_{j+l=i+1} H_j(B^{k+1}, B^k) \tensor H_l(F) - \end{align*} - \todo{Write down actual kunneth theorem with torsion (still works though)} - Note that this is the graded tensor product, and that the term $H_{i+1}(B^{k+1}, B^k) \tensor H_0(F) = H_{i+1}(B^{k+1}, B^k)$ and that this identification is compatible with the induced map $p_\ast : H_{i+1}(E^{k+1}, E^k) \to H_{i+1}(B^{k+1}, B^k)$ (hence the map is surjective). To prove that the map is a $\C$-iso, we need to prove that the kernel is in $\C$. The kernel is the sum of the following terms, with $1 \leq l \leq i+1$: - $$ H_{i+1-l}(B^{k+1}, B^k) \tensor H_l(F). $$ - Now we can use the assumption that $H_l(F) \in \C$ for $1 \leq l < n$ and that for $B \in \C$ we have $A \tensor B \in \C$ for all $A$ (by \LemmaRef{Serre-properties}). This concludes that the kernel $H_{i+1-l}(B^{k+1}, B^k) \tensor H_l(F)$ is indeed in $\C$. And hence the induced map is a $\C$-iso for all - $$ p_\ast : H_{i+1}(E^{k+1}, E^k) \to H_{i+1}(B^{k+1}, B^k) $$ - - This finished the induction on $k$. + Now we can apply the Künneth theorem for this product to obtain a natural short exact sequence, furthermore we apply the Künneth theorem for $(B^{k+1}, B^k) \times \ast$ to obtain a second short exact sequence as follows. + \[ \scriptsize \xymatrix @C=0.4cm { + 0 \ar[r] & (H(B^{k+1}, B^k) \tensor H(F))_{i+1} \ar[d]^-{p'} \ar[r] & H_{i+1}(E^{k+1}, E^k) \ar[d]^-{p_\ast} \ar[r] & \Tor(H(B^{k+1}, B^k), H(F))_i \ar[d]^-{p''} \ar[r] & 0 \\ + 0 \ar[r] & H_{i+1}(B^{k+1}, B^k) \ar[r] & H_{i+1}(B^{k+1}, B^k) \ar[r] & 0 \ar[r] & 0 + }\] + Now it remains to show that $p'$ and $p''$ are $\C$-iso, as it will then follow from the five lemma that $p_\ast$ is a $\C$-iso. + + First note that $p'$ is surjective as it is an isomorphism on the subspace $H_{i+1}(B^{k+1}, B^k) \tensor H_0(F)$. Its kernel on the other hand is precisely given by the terms $H_{i+1-q}(B^{k+1}, B^k) \tensor H_q(F)$ for $q>0$. By assumption we have $H_q(F) \in \C$ for all $0 < q < n$ and $H_{i+1}(B^{k+1}, B^k) = 0$ for all $i+1 \leq m$. By the tensor property of a Serre class the kernel is in $\C$ for all $i < n+m$. So indeed $p'$ is a $\C$-iso for all $i < n+m$. + + For $p''$ a similar reasoning holds, it is clearly surjective and we only need to prove that the kernel of $p''$ (which is the Tor group itself) is in $\C$. First notice that $\Tor(H_i(B^{k+1}, B^k), H_0(F)) = 0$ as $H_0(F) \iso \Z$. Then consider the other terms of the graded Tor group. Again we use the assumed bounds to conclude that the Tor group is in $\C$ for $i \leq n+m$. So indeed $p''$ is a $\C$-iso for all $i \leq n+m$. + + Now we conclude that $p_\ast : H_{i+1}(B^{k+1}, B^k) \to H_{i+1}(E^{k+1}, E^k)$ is indeed a $\C$-iso for all $i < n+m$. And by the long exact sequence of triples shown above we get a $\C$-iso $p_\ast : H_i(E^{k}, F) \to H_i(B^{k}, b_0)$ for all $i \leq n+m$. This finished the induction on $k$. + + This concludes that $H_i(E, F) \to H_i(B, b_0)$ is a $\C$-iso and by another application of the long exact sequence (of the pair $(E,F)$) and the five lemma we get the $\C$-iso $H_i(E) \to H_i(B)$. } \Lemma{homology-em-space}{