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If $X$ is a simply connected space and $\alpha\in\pi_n(X)$ and $Y=X\cup_{\alpha} e^{n+1}$. and $(\wedge V,d)$ be the minimal model for $X$,then $(\wedge V\oplus \mathbb{Q} u,d)$ is a commutative model for $Y$, which we denote by $M_\alpha$. If $f\colon X\rightarrow X_1$ and $Y=X\cup_{\alpha}e^{n+1}$ and $Y_1=X_1\cup_{f^*(\alpha)}e^{n+1}$,then we have a map $g\colon X_1\rightarrow Y_1$ and the following commutative diagram (1)

\begin{array}{lll} X & \rightarrow &Y\\ f\downarrow && \downarrow g\\ X_1 & \rightarrow & Y_1 \end{array}

Now,if $(\wedge V_1,d)$ is the minimal model for $X_1$ and so $(\wedge V_1\oplus \mathbb{Q}u,d)$ is a commutative model for $Y_1$ and if $\phi\colon (\wedge V_1,d)\rightarrow (\wedge V,d)$ be the Sullivan representative for $f$,then we have induced map $\psi$ from $(\wedge V_1\oplus\mathbb{Q}u)$ to $(\wedge V\oplus\mathbb{Q}u,d)$which is $\phi$ on $(\wedge V_1)$ and identity on $u$.

Then we have the following diagram (2)

\begin{array}{lll} (\wedge V_1\oplus\mathbb{Q} u,d) & \rightarrow &(\wedge V_1,d)\\ \psi\downarrow & & \downarrow \phi\\ (\wedge V\oplus\mathbb{Q}u,d) & \rightarrow &(\wedge V,d)\\ \end{array}

Now,my questions are

a. Is $\psi$ a commuatative model for $g\colon Y\rightarrow Y_1$?

b. Is (3) a commutative model for (2)?

If $X$ is a simply connected space and $\alpha\in\pi_n(X)$ and $Y=X\cup_{\alpha} e^{n+1}$. and $(\wedge V,d)$ be the minimal model for $X$,then $(\wedge V\oplus \mathbb{Q} u,d)$ is a commutative model for $Y$, which we denote by $M_\alpha$. If $f\colon X\rightarrow X_1$ and $Y=X\cup_{\alpha}e^{n+1}$ and $Y_1=X_1\cup_{f^*(\alpha)}e^{n+1}$,then we have a canonical map $g\colon X_1\rightarrow Y_1$ and the following commutative diagram (1)

\begin{array}{lll} X & \rightarrow &Y\\ f\downarrow && \downarrow g\\ X_1 & \rightarrow & Y_1 \end{array}

Now,if $(\wedge V_1,d)$ is the minimal model for $X_1$ and so $(\wedge V_1\oplus \mathbb{Q}u,d)$ is a commutative model for $Y_1$ and if $\phi\colon (\wedge V_1,d)\rightarrow (\wedge V,d)$ be the Sullivan representative for $f$,then we have induced map $\psi$ from $(\wedge V_1\oplus\mathbb{Q}u)$ to $(\wedge V\oplus\mathbb{Q}u,d)$which is $\phi$ on $(\wedge V_1)$ and identity on $u$.

Then we have the following diagram (2)

\begin{array}{lll} (\wedge V_1\oplus\mathbb{Q} u,d) & \rightarrow &(\wedge V_1,d)\\ \psi\downarrow & & \downarrow \phi\\ (\wedge V\oplus\mathbb{Q}u,d) & \rightarrow &(\wedge V,d)\\ \end{array}

Now,my questions are

a. Is $\psi$ a commuatative model for $g\colon Y\rightarrow Y_1$?

b. Is (2) a commutative model for (1)?

If $X$ is a simply connected space and $\alpha\in\pi_n(X)$ and $Y=X\cup_{\alpha} e^{n+1}$. and $(\wedge V,d)$ be the minimal model for $X$,then $(\wedge V\oplus \mathbb{Q} u,d)$ is a commutative model for $Y$, which we denote by $M_\alpha$. If $f\colon X\rightarrow X_1$ and $Y=X\cup_{\alpha}e^{n+1}$ and $Y_1=X_1\cup_{f^*(\alpha)}e^{n+1}$,then we have a map $g\colon X_1\rightarrow Y_1$ and the following commutative diagram (1)

\begin{array}{lll} X & \rightarrow &Y\\ f\downarrow && \downarrow g\\ X_1 & \rightarrow & Y_1 \end{array}

Now,if $(\wedge V_1,d)$ is the minimal model for $X_1$ and so $(\wedge V_1\oplus \mathbb{Q}u,d)$ is a commutative model for $Y_1$ and if $\phi\colon (\wedge V_1,d)\rightarrow (\wedge V,d)$ be the Sullivan representative for $f$,then we have induced map $\psi$ from $(\wedge V_1\oplus\mathbb{Q}u)$ to $(\wedge V\oplus\mathbb{Q}u,d)$which is $\phi$ on $(\wedge V_1)$ and identity on $u$.

Then we have the following diagram (2)---

\begin{array}{lll} (\wedge V_1\oplus\mathbb{Q} u,d) & \rightarrow (\wedge V_1,d)\\ \psi\downarrow & \downarrow \phi\\ (\wedge V\oplus\mathbb{Q}u,d) & \rightarrow (\wedge V,d)\\ \end{array}

Now,my questions are--

a. Is $\psi$ a commuatative model for $g\colon Y\rightarrow Y_1$?

b. Is (3) a commutative model for (2)?

If $X$ is a simply connected space and $\alpha\in\pi_n(X)$ and $Y=X\cup_{\alpha} e^{n+1}$. and $(\wedge V,d)$ be the minimal model for $X$,then $(\wedge V\oplus \mathbb{Q} u,d)$ is a commutative model for $Y$, which we denote by $M_\alpha$. If $f\colon X\rightarrow X_1$ and $Y=X\cup_{\alpha}e^{n+1}$ and $Y_1=X_1\cup_{f^*(\alpha)}e^{n+1}$,then we have a map $g\colon X_1\rightarrow Y_1$ and the following commutative diagram (1)

\begin{array}{lll} X & \rightarrow &Y\\ f\downarrow && \downarrow g\\ X_1 & \rightarrow & Y_1 \end{array}

Now,if $(\wedge V_1,d)$ is the minimal model for $X_1$ and so $(\wedge V_1\oplus \mathbb{Q}u,d)$ is a commutative model for $Y_1$ and if $\phi\colon (\wedge V_1,d)\rightarrow (\wedge V,d)$ be the Sullivan representative for $f$,then we have induced map $\psi$ from $(\wedge V_1\oplus\mathbb{Q}u)$ to $(\wedge V\oplus\mathbb{Q}u,d)$which is $\phi$ on $(\wedge V_1)$ and identity on $u$.

Then we have the following diagram (2)

\begin{array}{lll} (\wedge V_1\oplus\mathbb{Q} u,d) & \rightarrow &(\wedge V_1,d)\\ \psi\downarrow & & \downarrow \phi\\ (\wedge V\oplus\mathbb{Q}u,d) & \rightarrow &(\wedge V,d)\\ \end{array}

Now,my questions are

a. Is $\psi$ a commuatative model for $g\colon Y\rightarrow Y_1$?

b. Is (3) a commutative model for (2)?

If $X$ is a simply connected space and $\alpha\in\pi_n(X)$ and $Y=X\cup_{\alpha} e^{n+1}$. and $(\wedge V,d)$ be the minimal model for $X$,then $(\wedge V\oplus \mathbb{Q} u,d)$ is a commutative model for $Y$, which we denote by $M_\alpha$. If $f\colon X\rightarrow X_1$ and $Y=X\cup_{\alpha}e^{n+1}$ and $Y_1=X_1\cup_{f^*(\alpha)}e^{n+1}$,then we have a map $g\colon X_1\rightarrow Y_1$ and the following commutative diagram (1)

\begin{array}{lll} X \rightarrow Y\ f\downarrow \downarrow g\ X_1 \rightarrow Y_1

\end{array}

Now,if $(\wedge V_1,d)$ is the minimal model for $X_1$ and so $(\wedge V_1\oplus \mathbb{Q}u,d)$ is a commutative model for $Y_1$ and if $\phi\colon (\wedge V_1,d)\rightarrow (\wedge V,d)$ be the Sullivan representative for $f$,then we have induced map $\psi$ from $(\wedge V_1\oplus\mathbb{Q}u)$ to $(\wedge V\oplus\mathbb{Q}u,d)$which is $\phi$ on $(\wedge V_1)$ and identity on $u$.\ Then we have the following diagram (2)---\ [ \begin{array}{lll} (\wedge V_1\oplus\mathbb{Q} u,d) & \rightarrow (\wedge V_1,d)\\ \psi\downarrow & \downarrow \phi\\ (\wedge V\oplus\mathbb{Q}u,d) & \rightarrow (\wedge V,d)\\ \end{array} ]

Now,my questions are-- a)Is $\psi$ a commuatative model for $g\colon Y\rightarrow Y_1$? b)Is (3) a commutative model for (2)?

If $X$ is a simply connected space and $\alpha\in\pi_n(X)$ and $Y=X\cup_{\alpha} e^{n+1}$. and $(\wedge V,d)$ be the minimal model for $X$,then $(\wedge V\oplus \mathbb{Q} u,d)$ is a commutative model for $Y$, which we denote by $M_\alpha$. If $f\colon X\rightarrow X_1$ and $Y=X\cup_{\alpha}e^{n+1}$ and $Y_1=X_1\cup_{f^*(\alpha)}e^{n+1}$,then we have a map $g\colon X_1\rightarrow Y_1$ and the following commutative diagram (1)

\begin{array}{lll} X & \rightarrow &Y\\ f\downarrow && \downarrow g\\ X_1 & \rightarrow & Y_1 \end{array}

Now,if $(\wedge V_1,d)$ is the minimal model for $X_1$ and so $(\wedge V_1\oplus \mathbb{Q}u,d)$ is a commutative model for $Y_1$ and if $\phi\colon (\wedge V_1,d)\rightarrow (\wedge V,d)$ be the Sullivan representative for $f$,then we have induced map $\psi$ from $(\wedge V_1\oplus\mathbb{Q}u)$ to $(\wedge V\oplus\mathbb{Q}u,d)$which is $\phi$ on $(\wedge V_1)$ and identity on $u$.

Then we have the following diagram (2)--- 

\begin{array}{lll} (\wedge V_1\oplus\mathbb{Q} u,d) & \rightarrow (\wedge V_1,d)\\ \psi\downarrow & \downarrow \phi\\ (\wedge V\oplus\mathbb{Q}u,d) & \rightarrow (\wedge V,d)\\ \end{array}

Now,my questions are--

a. Is $\psi$ a commuatative model for $g\colon Y\rightarrow Y_1$?

b. Is (3) a commutative model for (2)?