Formalize pushout to map of homotopy groups
$begingroup$
Given a finite CW-complex $X$ and a space $Y$, together with two continuous maps $f, gcolon X to Y$, which agree on the $k$-th skeleton $X^k$ of $X$.
Given a $(k+1)$-cell of $X$ with characteristic map $e: (D^{k+1}, S^k) to (X^{k+1}, X^k)$ one can ask whether $f$ and $g$ are homotopic on $e$.
To do this, one can consider the pushout of two $D^{k+1}$ along $S^k$. Using the map $f circ e$ on one side and $g circ e$ on the other yields a unique map $S^{k+1} to Y$, i.e. a unique element of $pi_{k+1}(Y)$.
Both maps are homotopic on $e$ if and only if the given element is trivial in $pi_{k+1}(Y)$. No magic.
I would like to formalize this process as a map
$$C^{cell}_{k+1} cong pi_k(X^{k+1}, X^k) to pi_k(Y).$$
It is easy to see, that the element represented by the pushout does not depend on the homotopy class of $e$. This is what I have: let $P = f(X^k) = g(X^k)$, then
$$pi_k(X^{k+1}, X^k) xrightarrow{f_* - g_*} pi_k(Y, P)$$
because $f$ and $g$ agree on $X^k$ and the naturality of $delta$, the image of $f_* - g_*$ is in the kernel of $delta$, where $delta$ is the connecting homomorphism for the long exact homotopy sequence for the pair $(X^{k+1}, X^k)$ resp. $(Y, P)$.
Hence, for the image of a cell $e$, there exists and element in $pi_k(Y)$, which maps to the image of $e$. However, this element is only unique up to an element of $pi_k(P)$.
At this point I am missing some tools. I would expect that the "lift" is unique, but in general I can not assume, that the inclusion of $pi_k(P)$ is trivial. How do I overcome this ambiguity?
algebraic-topology
asked Dec 30 '18 at 15:34
lalamlalam
204
$endgroup$
add a comment |
$begingroup$
Given a finite CW-complex $X$ and a space $Y$, together with two continuous maps $f, gcolon X to Y$, which agree on the $k$-th skeleton $X^k$ of $X$.
Given a $(k+1)$-cell of $X$ with characteristic map $e: (D^{k+1}, S^k) to (X^{k+1}, X^k)$ one can ask whether $f$ and $g$ are homotopic on $e$.
To do this, one can consider the pushout of two $D^{k+1}$ along $S^k$. Using the map $f circ e$ on one side and $g circ e$ on the other yields a unique map $S^{k+1} to Y$, i.e. a unique element of $pi_{k+1}(Y)$.
Both maps are homotopic on $e$ if and only if the given element is trivial in $pi_{k+1}(Y)$. No magic.
I would like to formalize this process as a map
$$C^{cell}_{k+1} cong pi_k(X^{k+1}, X^k) to pi_k(Y).$$
It is easy to see, that the element represented by the pushout does not depend on the homotopy class of $e$. This is what I have: let $P = f(X^k) = g(X^k)$, then
$$pi_k(X^{k+1}, X^k) xrightarrow{f_* - g_*} pi_k(Y, P)$$
because $f$ and $g$ agree on $X^k$ and the naturality of $delta$, the image of $f_* - g_*$ is in the kernel of $delta$, where $delta$ is the connecting homomorphism for the long exact homotopy sequence for the pair $(X^{k+1}, X^k)$ resp. $(Y, P)$.
Hence, for the image of a cell $e$, there exists and element in $pi_k(Y)$, which maps to the image of $e$. However, this element is only unique up to an element of $pi_k(P)$.
At this point I am missing some tools. I would expect that the "lift" is unique, but in general I can not assume, that the inclusion of $pi_k(P)$ is trivial. How do I overcome this ambiguity?
algebraic-topology
asked Dec 30 '18 at 15:34
lalamlalam
204
$endgroup$
add a comment |
$begingroup$
Given a finite CW-complex $X$ and a space $Y$, together with two continuous maps $f, gcolon X to Y$, which agree on the $k$-th skeleton $X^k$ of $X$.
Given a $(k+1)$-cell of $X$ with characteristic map $e: (D^{k+1}, S^k) to (X^{k+1}, X^k)$ one can ask whether $f$ and $g$ are homotopic on $e$.
To do this, one can consider the pushout of two $D^{k+1}$ along $S^k$. Using the map $f circ e$ on one side and $g circ e$ on the other yields a unique map $S^{k+1} to Y$, i.e. a unique element of $pi_{k+1}(Y)$.
Both maps are homotopic on $e$ if and only if the given element is trivial in $pi_{k+1}(Y)$. No magic.
I would like to formalize this process as a map
$$C^{cell}_{k+1} cong pi_k(X^{k+1}, X^k) to pi_k(Y).$$
It is easy to see, that the element represented by the pushout does not depend on the homotopy class of $e$. This is what I have: let $P = f(X^k) = g(X^k)$, then
$$pi_k(X^{k+1}, X^k) xrightarrow{f_* - g_*} pi_k(Y, P)$$
because $f$ and $g$ agree on $X^k$ and the naturality of $delta$, the image of $f_* - g_*$ is in the kernel of $delta$, where $delta$ is the connecting homomorphism for the long exact homotopy sequence for the pair $(X^{k+1}, X^k)$ resp. $(Y, P)$.
Hence, for the image of a cell $e$, there exists and element in $pi_k(Y)$, which maps to the image of $e$. However, this element is only unique up to an element of $pi_k(P)$.
At this point I am missing some tools. I would expect that the "lift" is unique, but in general I can not assume, that the inclusion of $pi_k(P)$ is trivial. How do I overcome this ambiguity?
algebraic-topology
asked Dec 30 '18 at 15:34
lalamlalam
204
$endgroup$
Given a finite CW-complex $X$ and a space $Y$, together with two continuous maps $f, gcolon X to Y$, which agree on the $k$-th skeleton $X^k$ of $X$.
Given a $(k+1)$-cell of $X$ with characteristic map $e: (D^{k+1}, S^k) to (X^{k+1}, X^k)$ one can ask whether $f$ and $g$ are homotopic on $e$.
To do this, one can consider the pushout of two $D^{k+1}$ along $S^k$. Using the map $f circ e$ on one side and $g circ e$ on the other yields a unique map $S^{k+1} to Y$, i.e. a unique element of $pi_{k+1}(Y)$.
Both maps are homotopic on $e$ if and only if the given element is trivial in $pi_{k+1}(Y)$. No magic.
I would like to formalize this process as a map
$$C^{cell}_{k+1} cong pi_k(X^{k+1}, X^k) to pi_k(Y).$$
It is easy to see, that the element represented by the pushout does not depend on the homotopy class of $e$. This is what I have: let $P = f(X^k) = g(X^k)$, then
$$pi_k(X^{k+1}, X^k) xrightarrow{f_* - g_*} pi_k(Y, P)$$
because $f$ and $g$ agree on $X^k$ and the naturality of $delta$, the image of $f_* - g_*$ is in the kernel of $delta$, where $delta$ is the connecting homomorphism for the long exact homotopy sequence for the pair $(X^{k+1}, X^k)$ resp. $(Y, P)$.
Hence, for the image of a cell $e$, there exists and element in $pi_k(Y)$, which maps to the image of $e$. However, this element is only unique up to an element of $pi_k(P)$.
At this point I am missing some tools. I would expect that the "lift" is unique, but in general I can not assume, that the inclusion of $pi_k(P)$ is trivial. How do I overcome this ambiguity?
algebraic-topology
algebraic-topology
asked Dec 30 '18 at 15:34
lalamlalam
204
asked Dec 30 '18 at 15:34
lalamlalam
204
asked Dec 30 '18 at 15:34
lalamlalam
204
asked Dec 30 '18 at 15:34
lalamlalam
204
asked Dec 30 '18 at 15:34
lalamlalam
204
204
add a comment |
add a comment |
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