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I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\mathbb{Z}/2)\to \pi_2^{s}(X)\to H_2(X,\mathbb{Z})\to 0$.

Question: Does it always split?

Edit: I've been working on a solution to my question, and I believe I've made progress. However, I've hit a roadblock in the proof and I'm hoping for some assistance. I will award the best answer to whoever can help me finish my proof or provide insight into my original question.

I think my exact sequence splits canonically! My idea is to show the following dual exact sequence splits canonically: $$ 0\to H^2(X,\mathbb{Q}/\mathbb{Z})\to\mathrm{Hom}_{\mathbb{Z}}(\pi_2^s(X),\mathbb{Q}/\mathbb{Z})\stackrel{h}{\to} H^1(X,\mathbb{Z}/2)\to 0, $$ where I use the canonical isomorphism $\mathrm{Hom}_{\mathbb{Z}}(H_1(X,\mathbb{Z}/2),\mathbb{Q}/\mathbb{Z})\cong H^1(X,\mathbb{Z}/2)$. Next, I define $\phi$ as the generator of $\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(K(\mathbb{Z}/2,1)),\mathbb{Q}/\mathbb{Z})\simeq\mathbb{Z}/2$. For each $f\in H^1(X,\mathbb{Z}/2)\cong[X,K(\mathbb{Z}/2,1)]$, I consider the pullback $f^*\phi\in\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$. This defines a map $[\phi]:H^1(X,\mathbb{Z}/2)\mapsto\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$ that sends $f$ to $f^*\phi$.

I'm tempted to believe that $[\phi]$ is a homomorphism and $[\phi]\circ h=\mathrm{id}_{H^1(X,\mathbb{Z}/2)}$, but I cannot come up with a proof. If you have any insights into this, I would greatly appreciate it. Thank you!

I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\mathbb{Z}/2)\to \pi_2^{s}(X)\to H_2(X,\mathbb{Z})\to 0$.

Question: Does it always split?

Edit: I've been working on a solution to my question, and I believe I've made progress. However, I've hit a roadblock in the proof and I'm hoping for some assistance. I will award the best answer to whoever can help me finish my proof or provide insight into my original question.

I think my exact sequence splits canonically! My idea is to show the following dual exact sequence splits canonically: $$ 0\to H^2(X,\mathbb{Q}/\mathbb{Z})\to\mathrm{Hom}_{\mathbb{Z}}(\pi_2^s(X),\mathbb{Q}/\mathbb{Z})\stackrel{h}{\to} H^1(X,\mathbb{Z}/2)\to 0, $$ where I use the canonical isomorphism $\mathrm{Hom}_{\mathbb{Z}}(H_1(X,\mathbb{Z}/2),\mathbb{Q}/\mathbb{Z})\cong H^1(X,\mathbb{Z}/2)$. Next, I define $\phi$ as the generator of $\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(K(\mathbb{Z}/2,1)),\mathbb{Q}/\mathbb{Z})\simeq\mathbb{Z}/2$. For each $f\in H^1(X,\mathbb{Z}/2)\cong[X,K(\mathbb{Z}/2,1)]$, I consider the pullback $f^*\phi\in\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$. This defines a map $[\phi]:H^1(X,\mathbb{Z}/2)\mapsto\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$ that sends $f$ to $f^*\phi$.

I'm tempted to believe that $[\phi]$ is a homomorphism and $[\phi]\circ h=\mathrm{id}_{H^1(X,\mathbb{Z}/2)}$, but I cannot come up with a proof. If you have any insights into this, I would greatly appreciate it. Thank you!

Edit2 I’m really happy to see some of you enjoyed this question! Then what’s left for myself is to figure out how my argument fails, especially the naturality part.

I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\mathbb{Z}/2)\to \pi_2^{s}(X)\to H_2(X,\mathbb{Z})\to 0$.

Question: Does it always split?

Edit: I've been working on a solution to my question, and I believe I've made progress. However, I've hit a roadblock in the proof and I'm hoping for some assistance. I will award the best answer to whoever can help me finish my proof or provide insight into my original question.

I think my exact sequence splits canonically! My idea is to show the following dual exact sequence splits canonically: $$ 0\to\mathrm{Hom}_{\mathbb{Z}}(H_2(X,\mathbb{Z}),\mathbb{Q}/\mathbb{Z})\to\mathrm{Hom}_{\mathbb{Z}}(\pi_2^s(X),\mathbb{Q}/\mathbb{Z})\stackrel{h}{\to} H^1(X,\mathbb{Z}/2)\to 0, $$ where I use the canonical isomorphism $\mathrm{Hom}_{\mathbb{Z}}(H_1(X,\mathbb{Z}/2),\mathbb{Q}/\mathbb{Z})\cong H^1(X,\mathbb{Z}/2)$. Next, I define $\phi$ as the generator of $\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(K(\mathbb{Z}/2,1)),\mathbb{Q}/\mathbb{Z})\simeq\mathbb{Z}/2$. For each $f\in H^1(X,\mathbb{Z}/2)\cong[X,K(\mathbb{Z}/2,1)]$, I consider the pullback $f^*\phi\in\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$. This defines a map $[\phi]:H^1(X,\mathbb{Z}/2)\mapsto\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$ that sends $f$ to $f^*\phi$.

I'm tempted to believe that $[\phi]$ is a homomorphism and $[\phi]\circ h=\mathrm{id}_{H^1(X,\mathbb{Z}/2)}$, but I cannot come up with a proof. If you have any insights into this, I would greatly appreciate it. Thank you!

I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\mathbb{Z}/2)\to \pi_2^{s}(X)\to H_2(X,\mathbb{Z})\to 0$.

Question: Does it always split?

Edit: I've been working on a solution to my question, and I believe I've made progress. However, I've hit a roadblock in the proof and I'm hoping for some assistance. I will award the best answer to whoever can help me finish my proof or provide insight into my original question.

I think my exact sequence splits canonically! My idea is to show the following dual exact sequence splits canonically: $$ 0\to H^2(X,\mathbb{Q}/\mathbb{Z})\to\mathrm{Hom}_{\mathbb{Z}}(\pi_2^s(X),\mathbb{Q}/\mathbb{Z})\stackrel{h}{\to} H^1(X,\mathbb{Z}/2)\to 0, $$ where I use the canonical isomorphism $\mathrm{Hom}_{\mathbb{Z}}(H_1(X,\mathbb{Z}/2),\mathbb{Q}/\mathbb{Z})\cong H^1(X,\mathbb{Z}/2)$. Next, I define $\phi$ as the generator of $\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(K(\mathbb{Z}/2,1)),\mathbb{Q}/\mathbb{Z})\simeq\mathbb{Z}/2$. For each $f\in H^1(X,\mathbb{Z}/2)\cong[X,K(\mathbb{Z}/2,1)]$, I consider the pullback $f^*\phi\in\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$. This defines a map $[\phi]:H^1(X,\mathbb{Z}/2)\mapsto\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$ that sends $f$ to $f^*\phi$.

I'm tempted to believe that $[\phi]$ is a homomorphism and $[\phi]\circ h=\mathrm{id}_{H^1(X,\mathbb{Z}/2)}$, but I cannot come up with a proof. If you have any insights into this, I would greatly appreciate it. Thank you!

I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\mathbb{Z}/2)\to \pi_2^{s}(X)\to H_2(X,\mathbb{Z})\to 0$.

Question: Does it always split?

If it splits, how can we prove it? (Does the Adams spectral sequence provide any insight?)

If it doesn't split, what is a simple counter-example? (Is some $BO(n)$ or $BO$ such a counter-example, i.e., $\pi_2^s(BO(n))\stackrel{?}{=}\mathbb{Z}/4$?)

I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\mathbb{Z}/2)\to \pi_2^{s}(X)\to H_2(X,\mathbb{Z})\to 0$.

Question: Does it always split?

Edit: I've been working on a solution to my question, and I believe I've made progress. However, I've hit a roadblock in the proof and I'm hoping for some assistance. I will award the best answer to whoever can help me finish my proof or provide insight into my original question.

I think my exact sequence splits canonically! My idea is to show the following dual exact sequence splits canonically: $$ 0\to\mathrm{Hom}_{\mathbb{Z}}(H_2(X,\mathbb{Z}),\mathbb{Q}/\mathbb{Z})\to\mathrm{Hom}_{\mathbb{Z}}(\pi_2^s(X),\mathbb{Q}/\mathbb{Z})\stackrel{h}{\to} H^1(X,\mathbb{Z}/2)\to 0, $$ where I use the canonical isomorphism $\mathrm{Hom}_{\mathbb{Z}}(H_1(X,\mathbb{Z}/2),\mathbb{Q}/\mathbb{Z})\cong H^1(X,\mathbb{Z}/2)$. Next, I define $\phi$ as the generator of $\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(K(\mathbb{Z}/2,1)),\mathbb{Q}/\mathbb{Z})\simeq\mathbb{Z}/2$. For each $f\in H^1(X,\mathbb{Z}/2)\cong[X,K(\mathbb{Z}/2,1)]$, I consider the pullback $f^*\phi\in\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$. This defines a map $[\phi]:H^1(X,\mathbb{Z}/2)\mapsto\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$ that sends $f$ to $f^*\phi$.

I'm tempted to believe that $[\phi]$ is a homomorphism and $[\phi]\circ h=\mathrm{id}_{H^1(X,\mathbb{Z}/2)}$, but I cannot come up with a proof. If you have any insights into this, I would greatly appreciate it. Thank you!