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John Klein
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The following might help answer the last part of your post:

In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For example, suppose we start with a fibration $F \to E \to B$, in which $B$ is connected and based. Then we have a map $$ E/F \to B $$ and Ganea computed its homotopy fiber as the topological join $F*\Omega B$ (which by the way can be identified with $F\wedge \Sigma\Omega B$ once a basepoint in $F$ is chosen). In particular, we see that if $E \to B$ is $r$-connected and $B$ is $s$-connected, then the map $E/F \to B$ is $(r+s+1)$-connected.

There is no reason to stop at this point since we can iterate the above: let's set $B_1 = E/F$. Then the homotopy fiber of the map $B_1 \to B$ is given by $F \ast \Omega B \ast \Omega B \simeq F \wedge \Sigma \Omega B \wedge \Sigma \Omega B$.

This procedure gives a "filtration" $$E := B_0 \to B_1 \to B_2 \to \cdots $$ of spaces over $B$, where $B_j \to B$ is $(r+j)$-connected and whose homotopy fiber is of the form $F \ast (\Omega B)^{\ast j}$.

Set $B_\infty := \text{hocolim}_j B_j$. Then the map $B_\infty \to B$ is a weak homotopy equivalence and we have provided a "filtration" $\{B_j\}$ of $B$ in which $B_0 = E$ and in which homotopy fibers of $B_j \to B$ are identified as $F$ with $j$ copies of $\Omega B$ joined on.

What is interesting here is that although the homotopy type of $B_j$ depends on that of $E = B_0$, the homotopy fibers of the map $B_j \to B$ only depend on $F$ and $B$ and are independent of which fibration $E \to B$ with fiber $F$ you start with.

Example: the universal bundle $EG \to BG$ of a topological group $G$. In this case the homotopy fiber of $B_j \to B$ $(B = BG)$ has the homotopy type of $j$-fold join of copies of $G$. The space $B_j$ is the orbits of $G$ on this iterated join, and we get the standard $j$-th filtration term in the bar construction of $G$.

A dual version: Ganea also gave a Hilton-Eckmann dual of the above theory, where we now start with a cofibration sequence $A\to X \to X/A$ and take the homotopy fiber $A_1$ of the map $X \to X/A$. This gives a map $A \to A_1$ The homotopy cofiber of this map can be identified (this time only in a range of dimensions) in terms of $A$ and $X/A$ (you might call it the ``co-join'' of $\Sigma A$ with $X/A$). This produces a tower of fibrations $$\cdots \to A_3\to A_2 \to A_1$$ and compatible maps $A \to A_j$ such that $A \to \lim_j A_j$ will be a weak equivalence under mild connectivity assumptions on $A$ and $A \to X$.

Remark on fiber/cofiber sequences: The following is my take on Tillman's answer. IMO, Algebraic Topologists have been historically a bit sloppy with the definitions (I have also been guilty of such sloppiness). For me, a cofiber sequence $A \to X \to C$ is really shorthand notation for a commutative homotopy cocartesian square $$ \require{AMScd} \begin{CD} A @>>> P \\ @VVV @VVV \\ X @>>> C \end{CD} $$ in which $P$ is some (possibly weakly) contractible space. Likewise a fiber sequence $F \to E \to B$ is shorthand notation for a commutative homotopy cartesian square $$ \require{AMScd} \begin{CD} F @>>> E \\ @VVV @VVV \\ P @>>> B \end{CD} $$ for some (possibly weakly) contractible space $P$. In the spectrum case, these two notions agree.

The following might help answer the last part of your post:

In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For example, suppose we start with a fibration $F \to E \to B$, in which $B$ is connected and based. Then we have a map $$ E/F \to B $$ and Ganea computed its homotopy fiber as the topological join $F*\Omega B$ (which by the way can be identified with $F\wedge \Sigma\Omega B$ once a basepoint in $F$ is chosen). In particular, we see that if $E \to B$ is $r$-connected and $B$ is $s$-connected, then the map $E/F \to B$ is $(r+s+1)$-connected.

There is no reason to stop at this point since we can iterate the above: let's set $B_1 = E/F$. Then the homotopy fiber of the map $B_1 \to B$ is given by $F \ast \Omega B \ast \Omega B \simeq F \wedge \Sigma \Omega B \wedge \Sigma \Omega B$.

This procedure gives a "filtration" $$E := B_0 \to B_1 \to B_2 \to \cdots $$ of spaces over $B$, where $B_j \to B$ is $(r+j)$-connected and whose homotopy fiber is of the form $F \ast (\Omega B)^{\ast j}$.

Set $B_\infty := \text{hocolim}_j B_j$. Then the map $B_\infty \to B$ is a weak homotopy equivalence and we have provided a "filtration" $\{B_j\}$ of $B$ in which $B_0 = E$ and in which homotopy fibers of $B_j \to B$ are identified as $F$ with $j$ copies of $\Omega B$ joined on.

What is interesting here is that although the homotopy type of $B_j$ depends on that of $E = B_0$, the homotopy fibers of the map $B_j \to B$ only depend on $F$ and $B$ and are independent of which fibration $E \to B$ with fiber $F$ you start with.

Example: the universal bundle $EG \to BG$ of a topological group $G$. In this case the homotopy fiber of $B_j \to B$ $(B = BG)$ has the homotopy type of $j$-fold join of copies of $G$. The space $B_j$ is the orbits of $G$ on this iterated join, and we get the standard $j$-th filtration term in the bar construction of $G$.

A dual version: Ganea also gave a Hilton-Eckmann dual of the above theory, where we now start with a cofibration sequence $A\to X \to X/A$ and take the homotopy fiber $A_1$ of the map $X \to X/A$. This gives a map $A \to A_1$ The homotopy cofiber of this map can be identified (this time only in a range of dimensions) in terms of $A$ and $X/A$ (you might call it the ``co-join'' of $\Sigma A$ with $X/A$). This produces a tower of fibrations $$\cdots \to A_3\to A_2 \to A_1$$ and compatible maps $A \to A_j$ such that $A \to \lim_j A_j$ will be a weak equivalence under mild connectivity assumptions on $A$ and $A \to X$.

The following might help answer the last part of your post:

In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For example, suppose we start with a fibration $F \to E \to B$, in which $B$ is connected and based. Then we have a map $$ E/F \to B $$ and Ganea computed its homotopy fiber as the topological join $F*\Omega B$ (which by the way can be identified with $F\wedge \Sigma\Omega B$ once a basepoint in $F$ is chosen). In particular, we see that if $E \to B$ is $r$-connected and $B$ is $s$-connected, then the map $E/F \to B$ is $(r+s+1)$-connected.

There is no reason to stop at this point since we can iterate the above: let's set $B_1 = E/F$. Then the homotopy fiber of the map $B_1 \to B$ is given by $F \ast \Omega B \ast \Omega B \simeq F \wedge \Sigma \Omega B \wedge \Sigma \Omega B$.

This procedure gives a "filtration" $$E := B_0 \to B_1 \to B_2 \to \cdots $$ of spaces over $B$, where $B_j \to B$ is $(r+j)$-connected and whose homotopy fiber is of the form $F \ast (\Omega B)^{\ast j}$.

Set $B_\infty := \text{hocolim}_j B_j$. Then the map $B_\infty \to B$ is a weak homotopy equivalence and we have provided a "filtration" $\{B_j\}$ of $B$ in which $B_0 = E$ and in which homotopy fibers of $B_j \to B$ are identified as $F$ with $j$ copies of $\Omega B$ joined on.

What is interesting here is that although the homotopy type of $B_j$ depends on that of $E = B_0$, the homotopy fibers of the map $B_j \to B$ only depend on $F$ and $B$ and are independent of which fibration $E \to B$ with fiber $F$ you start with.

Example: the universal bundle $EG \to BG$ of a topological group $G$. In this case the homotopy fiber of $B_j \to B$ $(B = BG)$ has the homotopy type of $j$-fold join of copies of $G$. The space $B_j$ is the orbits of $G$ on this iterated join, and we get the standard $j$-th filtration term in the bar construction of $G$.

A dual version: Ganea also gave a Hilton-Eckmann dual of the above theory, where we now start with a cofibration sequence $A\to X \to X/A$ and take the homotopy fiber $A_1$ of the map $X \to X/A$. This gives a map $A \to A_1$ The homotopy cofiber of this map can be identified (this time only in a range of dimensions) in terms of $A$ and $X/A$ (you might call it the ``co-join'' of $\Sigma A$ with $X/A$). This produces a tower of fibrations $$\cdots \to A_3\to A_2 \to A_1$$ and compatible maps $A \to A_j$ such that $A \to \lim_j A_j$ will be a weak equivalence under mild connectivity assumptions on $A$ and $A \to X$.

Remark on fiber/cofiber sequences: The following is my take on Tillman's answer. IMO, Algebraic Topologists have been historically a bit sloppy with the definitions (I have also been guilty of such sloppiness). For me, a cofiber sequence $A \to X \to C$ is really shorthand notation for a commutative homotopy cocartesian square $$ \require{AMScd} \begin{CD} A @>>> P \\ @VVV @VVV \\ X @>>> C \end{CD} $$ in which $P$ is some (possibly weakly) contractible space. Likewise a fiber sequence $F \to E \to B$ is shorthand notation for a commutative homotopy cartesian square $$ \require{AMScd} \begin{CD} F @>>> E \\ @VVV @VVV \\ P @>>> B \end{CD} $$ for some (possibly weakly) contractible space $P$. In the spectrum case, these two notions agree.

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John Klein
  • 18.8k
  • 53
  • 109

The following might help answer the last part of your post:

In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For example, suppose we start with a fibration $F \to E \to B$, in which $B$ is connected and based. Then we have a map $$ E/F \to B $$ and Ganea computed its homotopy fiber as the topological join $F*\Omega B$ (which by the way can be identified with $F\wedge \Sigma\Omega B$ once a basepoint in $F$ is chosen). In particular, we see that if $E \to B$ is $r$-connected and $B$ is $s$-connected, then the map $E/F \to B$ is $(r+s+1)$-connected.

There is no reason to stop at this point since we can iterate the above: let's set $B_1 = E/F$. Then the homotopy fiber of the map $B_1 \to B$ is given by $F \ast \Omega B \ast \Omega B \simeq F \wedge \Sigma \Omega B \wedge \Sigma \Omega B$.

This procedure gives a "filtration" $$E := B_0 \to B_1 \to B_2 \to \cdots $$ of spaces over $B$, where $B_j \to B$ is $(r+j)$-connected and whose homotopy fiber is of the form $F \ast (\Omega B)^{\ast j}$.

Set $B_\infty := \text{hocolim}_j B_j$. Then the map $B_\infty \to B$ is a weak homotopy equivalence and we have provided a "filtration" $\{B_j\}$ of $B$ in which $B_0 = E$ and in which homotopy fibers of $B_j \to B$ are identified as $F$ with with$j$ copies of $\Omega B$ joined on.

What is interesting here is that although the homotopy type of $B_j$ depends on that of $E = B_0$, the homotopy fibers of the map $B_j \to B$ only depend on $F$ and $B$ and are independent of which fibration $E \to B$ with fiber $F$ you start with.

Example: the universal bundle $EG \to BG$ of a topological group $G$. In this case the homotopy fiber of $B_j$$B_j \to B$ $(B = BG)$ has the homotopy type of orbits space of $G$ acting diagonally on the $j$-fold join of copies of $G$. This The space $B_j$ is the orbits of $G$ on this iterated join, and we get the standard $j$-th filtration term in the bar construction of $G$.

A dual version: Ganea also gave a Hilton-Eckmann dual of the above theory, where we now start with a cofibration sequence $A\to X \to X/A$ and take the homotopy fiber $A_1$ of the map $X \to X/A$. This gives a map $A \to A_1$ The homotopy cofiber of this map can be identified (this time only in a range of dimensions) in terms of $A$ and $X/A$ (you might call it the ``co-join'' of $\Sigma A$ with $X/A$). This produces a tower of fibrations $$\cdots \to A_3\to A_2 \to A_1$$ and compatible maps $A \to A_j$ such that $A \to \lim_j A_j$ will be a weak equivalence under mild connectivity assumptions on $A$ and $A \to X$.

The following might help answer the last part of your post:

In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For example, suppose we start with a fibration $F \to E \to B$, in which $B$ is connected and based. Then we have a map $$ E/F \to B $$ and Ganea computed its homotopy fiber as the topological join $F*\Omega B$ (which by the way can be identified with $F\wedge \Sigma\Omega B$ once a basepoint in $F$ is chosen). In particular, we see that if $E \to B$ is $r$-connected and $B$ is $s$-connected, then the map $E/F \to B$ is $(r+s+1)$-connected.

There is no reason to stop at this point since we can iterate the above: let's set $B_1 = E/F$. Then the homotopy fiber of the map $B_1 \to B$ is given by $F \ast \Omega B \ast \Omega B \simeq F \wedge \Sigma \Omega B \wedge \Sigma \Omega B$.

This procedure gives a "filtration" $$E := B_0 \to B_1 \to B_2 \to \cdots $$ of spaces over $B$, where $B_j \to B$ is $(r+j)$-connected and whose homotopy fiber is of the form $F \ast (\Omega B)^{\ast j}$.

Set $B_\infty := \text{hocolim}_j B_j$. Then the map $B_\infty \to B$ is a weak homotopy equivalence and we have provided a "filtration" $\{B_j\}$ of $B$ in which $B_0 = E$ and in which homotopy fibers of $B_j \to B$ are identified as $F$ with with copies of $\Omega B$ joined on.

What is interesting here is that although the homotopy type of $B_j$ depends on that of $E = B_0$, the homotopy fibers of the map $B_j \to B$ only depend on $F$ and $B$ and are independent of which fibration $E \to B$ with fiber $F$ you start with.

Example: the universal bundle $EG \to BG$ of a topological group $G$. In this case $B_j$ has the homotopy type of orbits space of $G$ acting diagonally on the $j$-fold join of copies of $G$. This is the $j$-th filtration term in the bar construction of $G$.

A dual version: Ganea also gave a Hilton-Eckmann dual of the above theory, where we now start with a cofibration sequence $A\to X \to X/A$ and take the homotopy fiber $A_1$ of the map $X \to X/A$. This gives a map $A \to A_1$ The homotopy cofiber of this map can be identified (this time only in a range of dimensions) in terms of $A$ and $X/A$ (you might call it the ``co-join'' of $\Sigma A$ with $X/A$). This produces a tower of fibrations $$\cdots \to A_3\to A_2 \to A_1$$ and compatible maps $A \to A_j$ such that $A \to \lim_j A_j$ will be a weak equivalence under mild connectivity assumptions on $A$ and $A \to X$.

The following might help answer the last part of your post:

In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For example, suppose we start with a fibration $F \to E \to B$, in which $B$ is connected and based. Then we have a map $$ E/F \to B $$ and Ganea computed its homotopy fiber as the topological join $F*\Omega B$ (which by the way can be identified with $F\wedge \Sigma\Omega B$ once a basepoint in $F$ is chosen). In particular, we see that if $E \to B$ is $r$-connected and $B$ is $s$-connected, then the map $E/F \to B$ is $(r+s+1)$-connected.

There is no reason to stop at this point since we can iterate the above: let's set $B_1 = E/F$. Then the homotopy fiber of the map $B_1 \to B$ is given by $F \ast \Omega B \ast \Omega B \simeq F \wedge \Sigma \Omega B \wedge \Sigma \Omega B$.

This procedure gives a "filtration" $$E := B_0 \to B_1 \to B_2 \to \cdots $$ of spaces over $B$, where $B_j \to B$ is $(r+j)$-connected and whose homotopy fiber is of the form $F \ast (\Omega B)^{\ast j}$.

Set $B_\infty := \text{hocolim}_j B_j$. Then the map $B_\infty \to B$ is a weak homotopy equivalence and we have provided a "filtration" $\{B_j\}$ of $B$ in which $B_0 = E$ and in which homotopy fibers of $B_j \to B$ are identified as $F$ with $j$ copies of $\Omega B$ joined on.

What is interesting here is that although the homotopy type of $B_j$ depends on that of $E = B_0$, the homotopy fibers of the map $B_j \to B$ only depend on $F$ and $B$ and are independent of which fibration $E \to B$ with fiber $F$ you start with.

Example: the universal bundle $EG \to BG$ of a topological group $G$. In this case the homotopy fiber of $B_j \to B$ $(B = BG)$ has the homotopy type of $j$-fold join of copies of $G$. The space $B_j$ is the orbits of $G$ on this iterated join, and we get the standard $j$-th filtration term in the bar construction of $G$.

A dual version: Ganea also gave a Hilton-Eckmann dual of the above theory, where we now start with a cofibration sequence $A\to X \to X/A$ and take the homotopy fiber $A_1$ of the map $X \to X/A$. This gives a map $A \to A_1$ The homotopy cofiber of this map can be identified (this time only in a range of dimensions) in terms of $A$ and $X/A$ (you might call it the ``co-join'' of $\Sigma A$ with $X/A$). This produces a tower of fibrations $$\cdots \to A_3\to A_2 \to A_1$$ and compatible maps $A \to A_j$ such that $A \to \lim_j A_j$ will be a weak equivalence under mild connectivity assumptions on $A$ and $A \to X$.

Source Link
John Klein
  • 18.8k
  • 53
  • 109

The following might help answer the last part of your post:

In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For example, suppose we start with a fibration $F \to E \to B$, in which $B$ is connected and based. Then we have a map $$ E/F \to B $$ and Ganea computed its homotopy fiber as the topological join $F*\Omega B$ (which by the way can be identified with $F\wedge \Sigma\Omega B$ once a basepoint in $F$ is chosen). In particular, we see that if $E \to B$ is $r$-connected and $B$ is $s$-connected, then the map $E/F \to B$ is $(r+s+1)$-connected.

There is no reason to stop at this point since we can iterate the above: let's set $B_1 = E/F$. Then the homotopy fiber of the map $B_1 \to B$ is given by $F \ast \Omega B \ast \Omega B \simeq F \wedge \Sigma \Omega B \wedge \Sigma \Omega B$.

This procedure gives a "filtration" $$E := B_0 \to B_1 \to B_2 \to \cdots $$ of spaces over $B$, where $B_j \to B$ is $(r+j)$-connected and whose homotopy fiber is of the form $F \ast (\Omega B)^{\ast j}$.

Set $B_\infty := \text{hocolim}_j B_j$. Then the map $B_\infty \to B$ is a weak homotopy equivalence and we have provided a "filtration" $\{B_j\}$ of $B$ in which $B_0 = E$ and in which homotopy fibers of $B_j \to B$ are identified as $F$ with with copies of $\Omega B$ joined on.

What is interesting here is that although the homotopy type of $B_j$ depends on that of $E = B_0$, the homotopy fibers of the map $B_j \to B$ only depend on $F$ and $B$ and are independent of which fibration $E \to B$ with fiber $F$ you start with.

Example: the universal bundle $EG \to BG$ of a topological group $G$. In this case $B_j$ has the homotopy type of orbits space of $G$ acting diagonally on the $j$-fold join of copies of $G$. This is the $j$-th filtration term in the bar construction of $G$.

A dual version: Ganea also gave a Hilton-Eckmann dual of the above theory, where we now start with a cofibration sequence $A\to X \to X/A$ and take the homotopy fiber $A_1$ of the map $X \to X/A$. This gives a map $A \to A_1$ The homotopy cofiber of this map can be identified (this time only in a range of dimensions) in terms of $A$ and $X/A$ (you might call it the ``co-join'' of $\Sigma A$ with $X/A$). This produces a tower of fibrations $$\cdots \to A_3\to A_2 \to A_1$$ and compatible maps $A \to A_j$ such that $A \to \lim_j A_j$ will be a weak equivalence under mild connectivity assumptions on $A$ and $A \to X$.