Connectivity of suspension-loop adjunction

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Let $X$ be a $k$-connected spectrum for $k in BbbZ$.



I want to deduce how connected the counit of $(Sigma^infty, Omega^infty)$- adjunction is, that is, how connected is the map
$$
Sigma^inftyOmega^infty X to X.
$$



Any help would be appreciated.










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  • For $kgeq 0$, it is $2k$-connected by Freudenthal's suspension theorem.
    – Fernando Muro
    Sep 5 at 11:07










  • @FernandoMuro I think you are slightly off.
    – John Klein
    Sep 5 at 11:31










  • @JohnKlein could be, but I don't think so. Freudenthat's suspension theorem shows that the homotopy groups of $Sigma^inftyOmega^infty X$ coincide up to dimension $2k$ with those of $Omega^infty X$, which in turn are those of $X$. Hence my comment.
    – Fernando Muro
    Sep 5 at 11:41











  • If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$ is $(2r+2)$-connected.
    – John Klein
    Sep 5 at 11:51






  • 2




    @JohnKlein sure, I didn't claim my bound was optimal, it's obviously not optimal for $k=0$ since $Omega^infty X$ has abelian fundamental group. Actually, my argument above shows that the map is $(2k+1)$-connected. Let me provide an alternative answer below with the same bound as you get.
    – Fernando Muro
    Sep 5 at 13:51














up vote
7
down vote

favorite
1












Let $X$ be a $k$-connected spectrum for $k in BbbZ$.



I want to deduce how connected the counit of $(Sigma^infty, Omega^infty)$- adjunction is, that is, how connected is the map
$$
Sigma^inftyOmega^infty X to X.
$$



Any help would be appreciated.










share|cite|improve this question























  • For $kgeq 0$, it is $2k$-connected by Freudenthal's suspension theorem.
    – Fernando Muro
    Sep 5 at 11:07










  • @FernandoMuro I think you are slightly off.
    – John Klein
    Sep 5 at 11:31










  • @JohnKlein could be, but I don't think so. Freudenthat's suspension theorem shows that the homotopy groups of $Sigma^inftyOmega^infty X$ coincide up to dimension $2k$ with those of $Omega^infty X$, which in turn are those of $X$. Hence my comment.
    – Fernando Muro
    Sep 5 at 11:41











  • If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$ is $(2r+2)$-connected.
    – John Klein
    Sep 5 at 11:51






  • 2




    @JohnKlein sure, I didn't claim my bound was optimal, it's obviously not optimal for $k=0$ since $Omega^infty X$ has abelian fundamental group. Actually, my argument above shows that the map is $(2k+1)$-connected. Let me provide an alternative answer below with the same bound as you get.
    – Fernando Muro
    Sep 5 at 13:51












up vote
7
down vote

favorite
1









up vote
7
down vote

favorite
1






1





Let $X$ be a $k$-connected spectrum for $k in BbbZ$.



I want to deduce how connected the counit of $(Sigma^infty, Omega^infty)$- adjunction is, that is, how connected is the map
$$
Sigma^inftyOmega^infty X to X.
$$



Any help would be appreciated.










share|cite|improve this question















Let $X$ be a $k$-connected spectrum for $k in BbbZ$.



I want to deduce how connected the counit of $(Sigma^infty, Omega^infty)$- adjunction is, that is, how connected is the map
$$
Sigma^inftyOmega^infty X to X.
$$



Any help would be appreciated.







at.algebraic-topology homotopy-theory stable-homotopy






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Sep 5 at 17:54









j.c.

10.7k33677




10.7k33677










asked Sep 5 at 10:36









274072

21217




21217











  • For $kgeq 0$, it is $2k$-connected by Freudenthal's suspension theorem.
    – Fernando Muro
    Sep 5 at 11:07










  • @FernandoMuro I think you are slightly off.
    – John Klein
    Sep 5 at 11:31










  • @JohnKlein could be, but I don't think so. Freudenthat's suspension theorem shows that the homotopy groups of $Sigma^inftyOmega^infty X$ coincide up to dimension $2k$ with those of $Omega^infty X$, which in turn are those of $X$. Hence my comment.
    – Fernando Muro
    Sep 5 at 11:41











  • If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$ is $(2r+2)$-connected.
    – John Klein
    Sep 5 at 11:51






  • 2




    @JohnKlein sure, I didn't claim my bound was optimal, it's obviously not optimal for $k=0$ since $Omega^infty X$ has abelian fundamental group. Actually, my argument above shows that the map is $(2k+1)$-connected. Let me provide an alternative answer below with the same bound as you get.
    – Fernando Muro
    Sep 5 at 13:51
















  • For $kgeq 0$, it is $2k$-connected by Freudenthal's suspension theorem.
    – Fernando Muro
    Sep 5 at 11:07










  • @FernandoMuro I think you are slightly off.
    – John Klein
    Sep 5 at 11:31










  • @JohnKlein could be, but I don't think so. Freudenthat's suspension theorem shows that the homotopy groups of $Sigma^inftyOmega^infty X$ coincide up to dimension $2k$ with those of $Omega^infty X$, which in turn are those of $X$. Hence my comment.
    – Fernando Muro
    Sep 5 at 11:41











  • If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$ is $(2r+2)$-connected.
    – John Klein
    Sep 5 at 11:51






  • 2




    @JohnKlein sure, I didn't claim my bound was optimal, it's obviously not optimal for $k=0$ since $Omega^infty X$ has abelian fundamental group. Actually, my argument above shows that the map is $(2k+1)$-connected. Let me provide an alternative answer below with the same bound as you get.
    – Fernando Muro
    Sep 5 at 13:51















For $kgeq 0$, it is $2k$-connected by Freudenthal's suspension theorem.
– Fernando Muro
Sep 5 at 11:07




For $kgeq 0$, it is $2k$-connected by Freudenthal's suspension theorem.
– Fernando Muro
Sep 5 at 11:07












@FernandoMuro I think you are slightly off.
– John Klein
Sep 5 at 11:31




@FernandoMuro I think you are slightly off.
– John Klein
Sep 5 at 11:31












@JohnKlein could be, but I don't think so. Freudenthat's suspension theorem shows that the homotopy groups of $Sigma^inftyOmega^infty X$ coincide up to dimension $2k$ with those of $Omega^infty X$, which in turn are those of $X$. Hence my comment.
– Fernando Muro
Sep 5 at 11:41





@JohnKlein could be, but I don't think so. Freudenthat's suspension theorem shows that the homotopy groups of $Sigma^inftyOmega^infty X$ coincide up to dimension $2k$ with those of $Omega^infty X$, which in turn are those of $X$. Hence my comment.
– Fernando Muro
Sep 5 at 11:41













If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$ is $(2r+2)$-connected.
– John Klein
Sep 5 at 11:51




If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$ is $(2r+2)$-connected.
– John Klein
Sep 5 at 11:51




2




2




@JohnKlein sure, I didn't claim my bound was optimal, it's obviously not optimal for $k=0$ since $Omega^infty X$ has abelian fundamental group. Actually, my argument above shows that the map is $(2k+1)$-connected. Let me provide an alternative answer below with the same bound as you get.
– Fernando Muro
Sep 5 at 13:51




@JohnKlein sure, I didn't claim my bound was optimal, it's obviously not optimal for $k=0$ since $Omega^infty X$ has abelian fundamental group. Actually, my argument above shows that the map is $(2k+1)$-connected. Let me provide an alternative answer below with the same bound as you get.
– Fernando Muro
Sep 5 at 13:51










2 Answers
2






active

oldest

votes

















up vote
10
down vote



accepted










If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$
is $(2r+2)$-connected.



Here's a sketch: apply the functor $Omega^infty$ to get the map of spaces
$$
Q(Omega^infty X) to Omega^infty X
$$
where $Q = Omega^inftySigma^infty$.
It will be enough to identify the connectivity of the latter.



This map has a section so it is surjective on homotopy in all degrees. In particular your map of spectra is surjective on homotopy in all degrees.



It's enough to compute the degree of injectivity.
Let $S^j to Q(Omega^infty X)$ be a map, $j le 2r+1$. Suppose that
its image in $pi_j(Omega^infty X)$ is trivial.



By Freudenthal, the map
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))
$$
is surjective in this degree ($j le 2r+1$). So our map desuspends to give a map
$S^j to Omega^infty X$. But the composite
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))to
pi_j(Omega^infty X)
$$
is the identity. This means that our desuspension is trivial, hence its iterated suspensions are as well.



Here is the essence of the above argument:
suppose $Ato Bto C$ are maps of based spaces in which $Ato B$ is $k$-connected and $Ato C$ is $(k+1)$-connected. Then $Bto C$ is $(k+1)$-connected as well. Apply this to
$$
Omega^infty X to Q(Omega^infty X) to Omega^infty X .
$$






share|cite|improve this answer


















  • 1




    Thanks for the answer John, it was very illuminating, and nice to be able to avoid using Snaith Splitting/ Taylor tower where possible.
    – 274072
    Sep 5 at 15:03

















up vote
9
down vote













Indeed, as John Klein shows, the map is $(2k+2)$-connected. Let me offer an alternative proof of the fact that, for $X$ a $k$-connective spectrum, $kgeq 0$, the homomorphism $pi_iSigma^inftyOmega^infty Xrightarrowpi_i X$ is an isomorphism for $ileq 2k+1$. As indicated by John, this homomorphism is surjective for all $i$ and it suffices to show that the canonical homomorphism $pi_iOmega^infty Xrightarrow pi_iSigma^inftyOmega^infty X$ is an iso for $ileq 2k+1$. We will actually prove that this is more generally true replacing $Omega^infty X$ with a $k$-connected space $Y$ with abelian fundamental group and vanishing Whitehead products, e.g. a single loop space. I though that someone might eventually find this useful, even if it is not what you're asking about.



The Freudenthal suspension theorem shows that, for an $k$-connected space $Y$, the suspension operator
[
Sigma_*colon pi_i Y longrightarrow pi_i+1Sigma Y
]
is an isomorphism for $ileq 2k$ and an epimorphism for $i=2k+1$. This and the fact that $Sigma^j Y$ is $(j+k)$-connected clearly shows that the natural map from the homotopy groups of $Y$ to those of its suspension spectrum $Sigma^infty Y$
[
pi_i Y longrightarrow pi_iSigma^infty Y,
]
which is a transfinite composition of suspension operators, is an iso for $ileq 2k$ and epi for $i=2k+1$.



Moreover, by the theorems of Blakers and Massey on the homotopy groups of triads, we have an exact sequence in the critical dimension
[
pi_k+1Yotimes pi_k+1Ystackrel[-,-]longrightarrow pi_2k+1YstackrelSigma_*twoheadrightarrow pi_2k+2Sigma Y
]
where the first arrow is the Whitehead product. This still makes sense for $k=0$ using the non-abelian tensor product of groups and the Brown-Loday non-abelian Van Kampen theorems. In this case $[-,-]$ is the commutator product.



If Whitehad products in $pi_*Y$ vanish and the fundamental group is abelian, then the suspension operator is also an isomorphism for $i=2k+1$, and we get an isomorphism between the homotopy groups of $Y$ and $Sigma^infty Y$ in this dimension too.






share|cite|improve this answer


















  • 1




    This is very cool, thanks!
    – 274072
    Sep 5 at 15:04










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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
10
down vote



accepted










If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$
is $(2r+2)$-connected.



Here's a sketch: apply the functor $Omega^infty$ to get the map of spaces
$$
Q(Omega^infty X) to Omega^infty X
$$
where $Q = Omega^inftySigma^infty$.
It will be enough to identify the connectivity of the latter.



This map has a section so it is surjective on homotopy in all degrees. In particular your map of spectra is surjective on homotopy in all degrees.



It's enough to compute the degree of injectivity.
Let $S^j to Q(Omega^infty X)$ be a map, $j le 2r+1$. Suppose that
its image in $pi_j(Omega^infty X)$ is trivial.



By Freudenthal, the map
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))
$$
is surjective in this degree ($j le 2r+1$). So our map desuspends to give a map
$S^j to Omega^infty X$. But the composite
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))to
pi_j(Omega^infty X)
$$
is the identity. This means that our desuspension is trivial, hence its iterated suspensions are as well.



Here is the essence of the above argument:
suppose $Ato Bto C$ are maps of based spaces in which $Ato B$ is $k$-connected and $Ato C$ is $(k+1)$-connected. Then $Bto C$ is $(k+1)$-connected as well. Apply this to
$$
Omega^infty X to Q(Omega^infty X) to Omega^infty X .
$$






share|cite|improve this answer


















  • 1




    Thanks for the answer John, it was very illuminating, and nice to be able to avoid using Snaith Splitting/ Taylor tower where possible.
    – 274072
    Sep 5 at 15:03














up vote
10
down vote



accepted










If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$
is $(2r+2)$-connected.



Here's a sketch: apply the functor $Omega^infty$ to get the map of spaces
$$
Q(Omega^infty X) to Omega^infty X
$$
where $Q = Omega^inftySigma^infty$.
It will be enough to identify the connectivity of the latter.



This map has a section so it is surjective on homotopy in all degrees. In particular your map of spectra is surjective on homotopy in all degrees.



It's enough to compute the degree of injectivity.
Let $S^j to Q(Omega^infty X)$ be a map, $j le 2r+1$. Suppose that
its image in $pi_j(Omega^infty X)$ is trivial.



By Freudenthal, the map
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))
$$
is surjective in this degree ($j le 2r+1$). So our map desuspends to give a map
$S^j to Omega^infty X$. But the composite
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))to
pi_j(Omega^infty X)
$$
is the identity. This means that our desuspension is trivial, hence its iterated suspensions are as well.



Here is the essence of the above argument:
suppose $Ato Bto C$ are maps of based spaces in which $Ato B$ is $k$-connected and $Ato C$ is $(k+1)$-connected. Then $Bto C$ is $(k+1)$-connected as well. Apply this to
$$
Omega^infty X to Q(Omega^infty X) to Omega^infty X .
$$






share|cite|improve this answer


















  • 1




    Thanks for the answer John, it was very illuminating, and nice to be able to avoid using Snaith Splitting/ Taylor tower where possible.
    – 274072
    Sep 5 at 15:03












up vote
10
down vote



accepted







up vote
10
down vote



accepted






If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$
is $(2r+2)$-connected.



Here's a sketch: apply the functor $Omega^infty$ to get the map of spaces
$$
Q(Omega^infty X) to Omega^infty X
$$
where $Q = Omega^inftySigma^infty$.
It will be enough to identify the connectivity of the latter.



This map has a section so it is surjective on homotopy in all degrees. In particular your map of spectra is surjective on homotopy in all degrees.



It's enough to compute the degree of injectivity.
Let $S^j to Q(Omega^infty X)$ be a map, $j le 2r+1$. Suppose that
its image in $pi_j(Omega^infty X)$ is trivial.



By Freudenthal, the map
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))
$$
is surjective in this degree ($j le 2r+1$). So our map desuspends to give a map
$S^j to Omega^infty X$. But the composite
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))to
pi_j(Omega^infty X)
$$
is the identity. This means that our desuspension is trivial, hence its iterated suspensions are as well.



Here is the essence of the above argument:
suppose $Ato Bto C$ are maps of based spaces in which $Ato B$ is $k$-connected and $Ato C$ is $(k+1)$-connected. Then $Bto C$ is $(k+1)$-connected as well. Apply this to
$$
Omega^infty X to Q(Omega^infty X) to Omega^infty X .
$$






share|cite|improve this answer














If the spectrum $X$ is $r$-connected, then the map $Sigma^inftyOmega^infty X to X$
is $(2r+2)$-connected.



Here's a sketch: apply the functor $Omega^infty$ to get the map of spaces
$$
Q(Omega^infty X) to Omega^infty X
$$
where $Q = Omega^inftySigma^infty$.
It will be enough to identify the connectivity of the latter.



This map has a section so it is surjective on homotopy in all degrees. In particular your map of spectra is surjective on homotopy in all degrees.



It's enough to compute the degree of injectivity.
Let $S^j to Q(Omega^infty X)$ be a map, $j le 2r+1$. Suppose that
its image in $pi_j(Omega^infty X)$ is trivial.



By Freudenthal, the map
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))
$$
is surjective in this degree ($j le 2r+1$). So our map desuspends to give a map
$S^j to Omega^infty X$. But the composite
$$
pi_j(Omega^infty X) to pi_j(Q(Omega^infty X))to
pi_j(Omega^infty X)
$$
is the identity. This means that our desuspension is trivial, hence its iterated suspensions are as well.



Here is the essence of the above argument:
suppose $Ato Bto C$ are maps of based spaces in which $Ato B$ is $k$-connected and $Ato C$ is $(k+1)$-connected. Then $Bto C$ is $(k+1)$-connected as well. Apply this to
$$
Omega^infty X to Q(Omega^infty X) to Omega^infty X .
$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Sep 5 at 19:00

























answered Sep 5 at 11:30









John Klein

11.3k3574




11.3k3574







  • 1




    Thanks for the answer John, it was very illuminating, and nice to be able to avoid using Snaith Splitting/ Taylor tower where possible.
    – 274072
    Sep 5 at 15:03












  • 1




    Thanks for the answer John, it was very illuminating, and nice to be able to avoid using Snaith Splitting/ Taylor tower where possible.
    – 274072
    Sep 5 at 15:03







1




1




Thanks for the answer John, it was very illuminating, and nice to be able to avoid using Snaith Splitting/ Taylor tower where possible.
– 274072
Sep 5 at 15:03




Thanks for the answer John, it was very illuminating, and nice to be able to avoid using Snaith Splitting/ Taylor tower where possible.
– 274072
Sep 5 at 15:03










up vote
9
down vote













Indeed, as John Klein shows, the map is $(2k+2)$-connected. Let me offer an alternative proof of the fact that, for $X$ a $k$-connective spectrum, $kgeq 0$, the homomorphism $pi_iSigma^inftyOmega^infty Xrightarrowpi_i X$ is an isomorphism for $ileq 2k+1$. As indicated by John, this homomorphism is surjective for all $i$ and it suffices to show that the canonical homomorphism $pi_iOmega^infty Xrightarrow pi_iSigma^inftyOmega^infty X$ is an iso for $ileq 2k+1$. We will actually prove that this is more generally true replacing $Omega^infty X$ with a $k$-connected space $Y$ with abelian fundamental group and vanishing Whitehead products, e.g. a single loop space. I though that someone might eventually find this useful, even if it is not what you're asking about.



The Freudenthal suspension theorem shows that, for an $k$-connected space $Y$, the suspension operator
[
Sigma_*colon pi_i Y longrightarrow pi_i+1Sigma Y
]
is an isomorphism for $ileq 2k$ and an epimorphism for $i=2k+1$. This and the fact that $Sigma^j Y$ is $(j+k)$-connected clearly shows that the natural map from the homotopy groups of $Y$ to those of its suspension spectrum $Sigma^infty Y$
[
pi_i Y longrightarrow pi_iSigma^infty Y,
]
which is a transfinite composition of suspension operators, is an iso for $ileq 2k$ and epi for $i=2k+1$.



Moreover, by the theorems of Blakers and Massey on the homotopy groups of triads, we have an exact sequence in the critical dimension
[
pi_k+1Yotimes pi_k+1Ystackrel[-,-]longrightarrow pi_2k+1YstackrelSigma_*twoheadrightarrow pi_2k+2Sigma Y
]
where the first arrow is the Whitehead product. This still makes sense for $k=0$ using the non-abelian tensor product of groups and the Brown-Loday non-abelian Van Kampen theorems. In this case $[-,-]$ is the commutator product.



If Whitehad products in $pi_*Y$ vanish and the fundamental group is abelian, then the suspension operator is also an isomorphism for $i=2k+1$, and we get an isomorphism between the homotopy groups of $Y$ and $Sigma^infty Y$ in this dimension too.






share|cite|improve this answer


















  • 1




    This is very cool, thanks!
    – 274072
    Sep 5 at 15:04














up vote
9
down vote













Indeed, as John Klein shows, the map is $(2k+2)$-connected. Let me offer an alternative proof of the fact that, for $X$ a $k$-connective spectrum, $kgeq 0$, the homomorphism $pi_iSigma^inftyOmega^infty Xrightarrowpi_i X$ is an isomorphism for $ileq 2k+1$. As indicated by John, this homomorphism is surjective for all $i$ and it suffices to show that the canonical homomorphism $pi_iOmega^infty Xrightarrow pi_iSigma^inftyOmega^infty X$ is an iso for $ileq 2k+1$. We will actually prove that this is more generally true replacing $Omega^infty X$ with a $k$-connected space $Y$ with abelian fundamental group and vanishing Whitehead products, e.g. a single loop space. I though that someone might eventually find this useful, even if it is not what you're asking about.



The Freudenthal suspension theorem shows that, for an $k$-connected space $Y$, the suspension operator
[
Sigma_*colon pi_i Y longrightarrow pi_i+1Sigma Y
]
is an isomorphism for $ileq 2k$ and an epimorphism for $i=2k+1$. This and the fact that $Sigma^j Y$ is $(j+k)$-connected clearly shows that the natural map from the homotopy groups of $Y$ to those of its suspension spectrum $Sigma^infty Y$
[
pi_i Y longrightarrow pi_iSigma^infty Y,
]
which is a transfinite composition of suspension operators, is an iso for $ileq 2k$ and epi for $i=2k+1$.



Moreover, by the theorems of Blakers and Massey on the homotopy groups of triads, we have an exact sequence in the critical dimension
[
pi_k+1Yotimes pi_k+1Ystackrel[-,-]longrightarrow pi_2k+1YstackrelSigma_*twoheadrightarrow pi_2k+2Sigma Y
]
where the first arrow is the Whitehead product. This still makes sense for $k=0$ using the non-abelian tensor product of groups and the Brown-Loday non-abelian Van Kampen theorems. In this case $[-,-]$ is the commutator product.



If Whitehad products in $pi_*Y$ vanish and the fundamental group is abelian, then the suspension operator is also an isomorphism for $i=2k+1$, and we get an isomorphism between the homotopy groups of $Y$ and $Sigma^infty Y$ in this dimension too.






share|cite|improve this answer


















  • 1




    This is very cool, thanks!
    – 274072
    Sep 5 at 15:04












up vote
9
down vote










up vote
9
down vote









Indeed, as John Klein shows, the map is $(2k+2)$-connected. Let me offer an alternative proof of the fact that, for $X$ a $k$-connective spectrum, $kgeq 0$, the homomorphism $pi_iSigma^inftyOmega^infty Xrightarrowpi_i X$ is an isomorphism for $ileq 2k+1$. As indicated by John, this homomorphism is surjective for all $i$ and it suffices to show that the canonical homomorphism $pi_iOmega^infty Xrightarrow pi_iSigma^inftyOmega^infty X$ is an iso for $ileq 2k+1$. We will actually prove that this is more generally true replacing $Omega^infty X$ with a $k$-connected space $Y$ with abelian fundamental group and vanishing Whitehead products, e.g. a single loop space. I though that someone might eventually find this useful, even if it is not what you're asking about.



The Freudenthal suspension theorem shows that, for an $k$-connected space $Y$, the suspension operator
[
Sigma_*colon pi_i Y longrightarrow pi_i+1Sigma Y
]
is an isomorphism for $ileq 2k$ and an epimorphism for $i=2k+1$. This and the fact that $Sigma^j Y$ is $(j+k)$-connected clearly shows that the natural map from the homotopy groups of $Y$ to those of its suspension spectrum $Sigma^infty Y$
[
pi_i Y longrightarrow pi_iSigma^infty Y,
]
which is a transfinite composition of suspension operators, is an iso for $ileq 2k$ and epi for $i=2k+1$.



Moreover, by the theorems of Blakers and Massey on the homotopy groups of triads, we have an exact sequence in the critical dimension
[
pi_k+1Yotimes pi_k+1Ystackrel[-,-]longrightarrow pi_2k+1YstackrelSigma_*twoheadrightarrow pi_2k+2Sigma Y
]
where the first arrow is the Whitehead product. This still makes sense for $k=0$ using the non-abelian tensor product of groups and the Brown-Loday non-abelian Van Kampen theorems. In this case $[-,-]$ is the commutator product.



If Whitehad products in $pi_*Y$ vanish and the fundamental group is abelian, then the suspension operator is also an isomorphism for $i=2k+1$, and we get an isomorphism between the homotopy groups of $Y$ and $Sigma^infty Y$ in this dimension too.






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Indeed, as John Klein shows, the map is $(2k+2)$-connected. Let me offer an alternative proof of the fact that, for $X$ a $k$-connective spectrum, $kgeq 0$, the homomorphism $pi_iSigma^inftyOmega^infty Xrightarrowpi_i X$ is an isomorphism for $ileq 2k+1$. As indicated by John, this homomorphism is surjective for all $i$ and it suffices to show that the canonical homomorphism $pi_iOmega^infty Xrightarrow pi_iSigma^inftyOmega^infty X$ is an iso for $ileq 2k+1$. We will actually prove that this is more generally true replacing $Omega^infty X$ with a $k$-connected space $Y$ with abelian fundamental group and vanishing Whitehead products, e.g. a single loop space. I though that someone might eventually find this useful, even if it is not what you're asking about.



The Freudenthal suspension theorem shows that, for an $k$-connected space $Y$, the suspension operator
[
Sigma_*colon pi_i Y longrightarrow pi_i+1Sigma Y
]
is an isomorphism for $ileq 2k$ and an epimorphism for $i=2k+1$. This and the fact that $Sigma^j Y$ is $(j+k)$-connected clearly shows that the natural map from the homotopy groups of $Y$ to those of its suspension spectrum $Sigma^infty Y$
[
pi_i Y longrightarrow pi_iSigma^infty Y,
]
which is a transfinite composition of suspension operators, is an iso for $ileq 2k$ and epi for $i=2k+1$.



Moreover, by the theorems of Blakers and Massey on the homotopy groups of triads, we have an exact sequence in the critical dimension
[
pi_k+1Yotimes pi_k+1Ystackrel[-,-]longrightarrow pi_2k+1YstackrelSigma_*twoheadrightarrow pi_2k+2Sigma Y
]
where the first arrow is the Whitehead product. This still makes sense for $k=0$ using the non-abelian tensor product of groups and the Brown-Loday non-abelian Van Kampen theorems. In this case $[-,-]$ is the commutator product.



If Whitehad products in $pi_*Y$ vanish and the fundamental group is abelian, then the suspension operator is also an isomorphism for $i=2k+1$, and we get an isomorphism between the homotopy groups of $Y$ and $Sigma^infty Y$ in this dimension too.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Sep 5 at 18:59

























answered Sep 5 at 14:26









Fernando Muro

11.3k13162




11.3k13162







  • 1




    This is very cool, thanks!
    – 274072
    Sep 5 at 15:04












  • 1




    This is very cool, thanks!
    – 274072
    Sep 5 at 15:04







1




1




This is very cool, thanks!
– 274072
Sep 5 at 15:04




This is very cool, thanks!
– 274072
Sep 5 at 15:04

















 

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