Joshua Moerman
10 years ago
13 changed files with 116 additions and 95 deletions
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\newcommand{\titleCDGA}{\texorpdfstring{$\CDGA_\k$}{CDGA}} |
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\section{Homotopy theory of \titleCDGA} |
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\label{sec:model-of-cdga} |
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\subsection{Cochain models for the $n$-disk and $n$-sphere} |
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\input{notes/CDGA_Basic_Examples} |
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\subsection{The Quillen model structure on \titleCDGA} |
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\input{notes/Model_Of_CDGA} |
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\subsection{Homotopy relations on \titleCDGA} |
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\input{notes/Homotopy_Relations_CDGA} |
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\section{Polynomial Forms} |
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\label{sec:cdga-of-polynomials} |
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\subsection{CDGA of Polynomials} |
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\input{notes/CDGA_Of_Polynomials} |
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\subsection{Polynomial Forms on a Space} |
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\label{sec:polynomial-forms} |
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\input{notes/Polynomial_Forms} |
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Although the abstract theory of model categories gives us tools to construct a homotopy relation (\DefinitionRef{homotopy}), it is useful to have a concrete notion of homotopic maps. |
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Consider the free cdga on one generator $\Lambda(t, dt)$, where $\deg{t} = 0$, this can be thought of as the (dual) unit interval with endpoints $1$ and $t$. We define two \emph{endpoint maps} as follows: |
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$$ d_0, d_1 : \Lambda(t, dt) \to \k $$ |
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$$ d_0(t) = 1, \qquad d_1(t) = 0, $$ |
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this extends linearly and multiplicatively. Note that it follows that we have $d_0(1-t) = 0$ and $d_1(1-t) = 1$. These two functions extend to tensorproducts as $d_0, d_1: \Lambda(t, dt) \tensor X \to \k \tensor X \tot{\iso} X$. |
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\Definition{cdga_homotopy}{ |
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We call $f, g: A \to X$ homotopic ($f \simeq g$) if there is a map |
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$$ h: A \to \Lambda(t, dt) \tensor X, $$ |
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such that $d_0 h = g$ and $d_1 h = f$. |
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} |
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In terms of model categories, such a homotopy is a right homotopy and the object $\Lambda(t, dt) \tensor X$ is a path object for $X$. We can easily see that it is a very good path object. First note that $\Lambda(t, dt) \tensor X \tot{(d_0, d_1)} X \oplus X$ is surjective (for $(x, y) \in X \oplus X$ take $t \tensor x + (1-t) \tensor y$). Secondly we note that $\Lambda(t, dt) = \Lambda(D(0))$ and hence $\k \to \Lambda(t, dt)$ is a cofibration, by \LemmaRef{model-cats-coproducts} we have that $X \to \Lambda(t, dt) \tensor X$ is a cofibration. |
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Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true. |
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\Lemma{cdga_homotopy}{ |
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If $A$ is a cofibrant cdga and $f \simeq^r g: A \to X$, then $f \simeq g$ in the above sense. |
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} |
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\Proof{ |
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Because $A$ is cofibrant, there is a very good homotopy $H$. Consider a lifting problem to construct a map $Path_X \to \Lambda(t, dt) \tensor X$. |
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} |
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\Corollary{cdga_homotopy_eqrel}{ |
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For cofibrant $A$, $\simeq$ defines a equivalence relation. |
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} |
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\Definition{cdga_homotopy_classes}{ |
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For cofibrant $A$ define the set of equivalence classes as: |
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$$ [A, X] = \Hom_{\CDGA_\k}(A, X) / \simeq. $$ |
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} |
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The results from model categories immediately imply the following results. |
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\Corollary{cdga_homotopy_properties}{ |
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Let $A$ be cofibrant. |
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\begin{itemize} |
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\item Let $i: A \to B$ be a trivial cofibration, then the induced map $i^\ast: [B, X] \to [A, X]$ is a bijection. |
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\item Let $p: X \to Y$ be a trivial fibration, then the induced map $p_\ast: [A, X] \to [A, Y]$ is a bijection. |
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\item Let $A$ and $X$ both be cofibrant, then $f: A \we X$ is a weak equivalence if and only if $f$ is a strong homotopy equivalence. Moreover, the two induced maps are bijections: |
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$$ f_\ast: [Z, A] \tot{\iso} [Z, X], $$ |
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$$ f^\ast: [X, Z] \tot{\iso} [A, X]. $$ |
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\end{itemize} |
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} |
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\Lemma{cdga_homotopy_homology}{ |
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Let $f, g: A \to X$ be two homotopic maps, then $H(f) = H(g): HA \to HX$. |
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} |
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\Proof{ |
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We only need to consider $H(d_0)$ and $H(d_1)$. |
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} |
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