Orbit spaces of involutions on spheres

I'm studying the following problem: Let

• $({\mathbb S}^N,\theta)$ be the $n$-sphere (in ${\mathbb R}^{N+1}$) endowed with the antipodal action $\theta:(x_0,\ldots,x_N)\to (-x_0,\ldots,-x_N)$;

• $({\mathbb S}^1,\tau)$ the $1$-sphere endowed with the antipodal action $\tau:(y_0,y_1)\to (y_0,-y_1)$.

On the cartesian product ${\mathbb S}^N\times {\mathbb S}^1$ I consider the diagonal action ${\rm T}:=\theta\times\tau$ in order to have a space with involution $({\mathbb S}^N\times {\mathbb S}^1,{\rm T})$.

I'm interesting in the orbit space $$Q^1_N\;:=\;{\mathbb S}^N\times {\mathbb S}^1\;/\;\sim_{\rm T}\;.$$ I read in some papers that $$Q^1_N\simeq {\mathbb R}P^{N+1}\;\sharp\; {\mathbb R}P^{N+1}\;,\qquad\; Q^1_\infty\simeq {\mathbb R}P^{\infty}\;\vee\; {\mathbb R}P^{\infty}\;,\qquad\quad {\rm (a)}$$ where in the first case $\sharp$ denotes the connected sum and in the case $N=\infty$ the symbol $\vee$ is the sum.

[1] Can someone explain me the proof of the identifications (a)?

Now, let me consider the torus ${\mathbb T}^d:={\mathbb S}^1\times\ldots\times{\mathbb S}^1$ with the (diagonal) involution $\tau\times\ldots\times\tau$ and the product space ${\mathbb S}^N\times {\mathbb T}^d$ with involution ${\rm T}:=\theta\times \tau\times\ldots\times\tau$. As before let $$Q^d_N\;:=\;{\mathbb S}^N\times {\mathbb S}^1\;/\;\sim_{\rm T}$$ be the orbit space.

[2] Is it possible to have a description for $Q^d_N$ of the type of the one in equation (a) when $d>1$? I am particularly interested to the cases $d=2,3$ and $N=\infty$.

Unfortunately I do not have much experience in topology and I really need these information to finish an important job. Thanks in advance, Giuseppe

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