Proof. The first statement follows directly from the uniqueness of tubular neighborhoods (see, e.g., [Reference Kosinski28, Corollary III.3.2]) and the second statement then follows from the fact that $\pi _1(\mathrm {SO}(n-1))\cong \mathbb {Z}/2$ , as $n\geq 5$ .

Proof. Let $\varphi _t$ and $\psi _t$ be extensions of $\varphi $ along F. Then the maps $[0,1]\times S^1\times D^{n-1}\to [0,1]\times M$ ,

$$\begin{align*}(t,\lambda,v)\mapsto (t,\varphi_t(\lambda,v)),\quad (t,\lambda,v)\mapsto (t,\psi_t(\lambda,v)) \end{align*}$$

are neat tubular neighborhoods of the neat submanifold $N=\{(t,F_t(\lambda ))\mid (t,\lambda )\in [0,1]\times S^1\}$ of $[0,1]\times M$ in the sense of [Reference Kosinski28, Chapter III.4]. By the uniqueness of neat tubular neighborhoods (see, e.g., [Reference Kosinski28, Theorem III.4.2 and subsequent remark]), after applying an isotopy of neat tubular neighborhoods that fixes N pointwise (which corresponds to isotopies of $\psi _0$ and $\psi _1$ on the boundary components that fix $f(S^1)$ pointwise), we can assume that

$$\begin{align*}\psi_t(\lambda,\cdot)=\varphi_t(\lambda,\alpha_{(t,\lambda)}(\cdot))\end{align*}$$

for a smooth map $\alpha \colon [0,1]\times S^1\to \mathrm {SO}(n-1)$ with $\alpha _{(0,\cdot )}$ homotopic to the constant map $\equiv \mathrm {id}$ . This shows that $\psi _1$ is equivalent to $\varphi _1$ , where the isotopy is given by $\varphi _1(\lambda ,\alpha _{(t,\lambda )}(\cdot ))$ .

Proof. Let $\xi =F_*[T^2]_{\mathbb {Z}/2}\in H_2(M;\mathbb {Z}/2)$ . By Proposition 2.10, the normal bundle $\nu _F$ is trivial if and only if $w_2(\nu _F)=0$ . We have

$$\begin{align*}F^*w_2(\nu_F)=F^*w_2(\nu_F\oplus TF(T^2))=F^*w_2(TM|_{F(T^2)})=w_2(F^*TM)=F^*w_2(M),\end{align*}$$

which vanishes if and only if

$$\begin{align*}0=F^*w_2(M)\frown [T^2]_{\mathbb{Z}/2}=w_2(M) \frown F_*[T^2]_{\mathbb{Z}/2}=w_2(M) \frown \xi. \end{align*}$$

Hence, $w_2(M)\frown \xi =0$ if and only if $\nu _F$ is trivial.

Now, suppose that $\nu _F$ is trivial, that is, there exists an embedding $\bar {F}\colon T^2\times D^{n-2}\hookrightarrow M$ such that $\bar {F}(\cdot ,0)=F$ . The map $\varphi _t\colon S^1 \times D^{n-1}\to M$ ,

$$\begin{align*}\varphi_t(\lambda,v)=\bar{F}((\lambda,e^{2\pi i(t+\frac{v_{n-1}}{4})}),(v_1,\dots,v_{n-2})) \end{align*}$$

for $\lambda \in S^1$ , $v=(v_1,\dots ,v_{n-1})\in D^{n-1}$ is an isotopy along $F_t$ with $\varphi _0=\varphi _1$ , showing that F induces the identity on equivalence classes of normal framings of f.

Finally, suppose that F induces the identity on equivalence classes of normal framings. Let $\varphi $ be a normal framing, and let $\varphi _t$ be an extension along $F_t$ . Then $\varphi =\varphi _0$ is equivalent to $\varphi _1$ . By modifying $\varphi _t$ for $t\in [1-\varepsilon ,1]$ for $\varepsilon>0$ small according to the isotopy between $\varphi $ and $\varphi _1$ , we can assume that $\varphi =\varphi _0=\varphi _1$ . Then we define the embedding $T^2\times D^{n-1}\hookrightarrow S^1\times M$ ,

$$\begin{align*}(\lambda,e^{2\pi i t},v)\mapsto(e^{2\pi i t},\varphi_t(\lambda,v)), \end{align*}$$

showing that the embedding $\overset {\circ }{F}\colon T^2\hookrightarrow S^1\times M$ , $(\lambda ,e^{2\pi i t})\mapsto (e^{2\pi i t},F_t(\lambda ))$ has trivial normal bundle. On the other hand, the normal bundle of $\overset {\circ }{F}(T^2)$ is isomorphic to the sum of the normal bundle $\nu _F$ of F in M with the trivial bundle $\underline {\mathbb {R}}_{T^2}$ (corresponding to paths of the form $s\mapsto (e^{2\pi i (t+s)},F_t(\lambda ))$ ). Thus, the bundle $\nu _F\oplus \underline {\mathbb {R}}_{T^2}$ is trivial. Since $n\geq 5$ , it follows from Proposition 2.10, together with the stability of the Stiefel–Whitney classes, that $\nu _F$ is trivial.

Proof. First, suppose that such a map h exists. By Theorem 2.8, we can assume that h is an embedding. Again, by Theorem 2.8, the map $T^2\to M,\, (\lambda _1,\lambda _2)\mapsto f(\lambda _1)$ , which induces the trivial map on $H_2$ , is homotopic to an embedding $F_0$ that extends f. Now, the connected sum embedding $T^2\# S^2\cong T^2\hookrightarrow M$ of $F_0$ and h satisfies the required properties.

Conversely, suppose that f is null-homotopic and that $F\colon T^2\hookrightarrow M$ is an embedding with $F(\cdot ,1)=f$ and $w_2(M)\frown F_*[T^2]_{\mathbb {Z}/2}\neq 0$ . Since f is null-homotopic, there exists an embedding $\bar {f}\colon D^2\hookrightarrow M$ with $\bar {f}|_{S^1}=f$ . We now define a map $\overline {F}\colon (D^2\times S^1)\setminus (D^3)^\circ \to M$ from the punctured solid torus to M by first defining it on $T^2\cup (D^2\times \{1\})$ (and assume that the deleted $D^3$ is disjoint from this part) by setting

$$\begin{align*}\overline{F}(x)=F(x)\text{ for }x\in T^2\text{ and }\overline{F}(y,1)=\bar{f}(y)\text{ for }y\in D^2. \end{align*}$$

Since $T^2\cup (D^2\times \{1\})$ is a deformation retract of $(D^2\times S^1)\setminus (D^3)^\circ $ , we can extend this map to all of $(D^2\times S^1)\setminus (D^3)^\circ $ and we have $\overline {F}|_{T^2}=F$ by construction. We define $h\colon S^2\to M$ as the restriction of $\overline {F}$ to the other boundary component. Since $T^2$ and $S^2$ define the same homology classes inside $(D^2\times S^1)\setminus (D^3)^\circ $ , it follows that they induce the same homology class $F_*[T^2]=h_*[S^2]$ .

Proof. Since, by Theorem 2.8, any map $S^1\to M$ is homotopic to an embedding, the map $\mu $ is surjective. Further, since any two homotopic embeddings $S^1\hookrightarrow M$ are isotopic by Theorem 2.9, the preimages of a map $f\colon S^1\to M$ under $\mu $ , which we can assume to be embeddings, can be represented by normal framings of f. This shows that there are either one or two preimages of the class represented by f, depending on whether the two nonequivalent normal framings of f are isotopic or not.

By Theorem 2.8, we can assume that every self-isotopy of f is an embedded torus. Hence, by Lemma 3.4, the two nonequivalent normal framings of f are isotopic if and only if there is an embedded torus $F\colon T^2\to M$ with $F(\cdot ,1)=f$ and $w_2(M)\frown F_*[T^2]_{\mathbb {Z}/2}\neq 0$ . In particular, if M is spin, then there is no such isotopy, which shows item (i). Further, by Lemma 3.5, a sufficient condition for the existence of such an embedding is the existence of a continuous map $h\colon S^2\to M$ with $w_2(M)\frown h_*[S^2]_{\mathbb {Z}/2}\neq 0$ , and this condition is also necessary if f is null-homotopic. We now show that this condition is satisfied on M if and only if it is satisfied on $\widetilde {M}$ , showing items (ii) and (iii).

For any map $h\colon S^2\to \widetilde {M}$ , we have

$$\begin{align*}w_2(M)\frown (\pi\circ h)_*[S^2]=w_2(M)\frown \pi_*h_*[S^2]=\pi_*(\pi^*w_2(M)\frown h_*[S^2])=\pi_*(w_2(\widetilde{M})\frown h_*[S^2]). \end{align*}$$

Since $\pi $ induces an isomorphism on $H_0$ , it follows that h satisfies the required property for $\widetilde {M}$ if and only if $\pi \circ h$ satisfies it for M. Since any map $S^2\to M$ can be lifted to $\widetilde {M}$ , the claim follows.

Proof. Since we can lift extensions to the universal cover, it follows that any lift of a trivial normal framing is trivial. Since, by Proposition 3.6, the numbers of isotopy classes of normal framings of null-homotopic embeddings for M and $\widetilde {M}$ coincide, it follows that a normal framing is trivial if and only if its lift to the universal cover is trivial.

Proof. By Lemma 2.4, since the inclusion of a fiber is null-homotopic, the pull-back of the Euler class of $\pi $ to $\widetilde {M}$ is primitive. This implies that the pull-back of P along the projection $\widetilde {M}\to M$ is simply connected by Lemma 2.3; in particular, it is the universal cover $\widetilde {P}$ , so we can write $\widetilde {P}\xrightarrow {\widetilde {\pi }}\widetilde {M}$ for the pull-back bundle. Hence, by Lemma 3.8, $\varphi _\pi $ is trivial if and only if $\varphi _{\widetilde {\pi }}$ is trivial, that is, we can assume that M and P are simply connected. This case now follows from [Reference Goldstein and Lininger17, Theorem 8].

Proof. Since f is null-homotopic, it bounds an embedded disc by Theorem 2.8. Then the statement of the lemma is well known. For completeness, we give the proof below.

We can write M as

$$ \begin{align*} M\cong M\# S^n\cong M\# (D^2\times S^{n-2}\cup_{\mathrm{id}_{S^1\times S^{n-2}}}S^1\times D^{n-1}) \end{align*} $$

and the inclusion $\varphi _0$ of $S^1\times D^{n-1}$ into the second factor is a trivial normal framing, the extension $\bar {\varphi }_0\colon D^2\times D^{n-2}\hookrightarrow (D^2\times S^{n-2})\cup _{S^1\times S^{n-2}} (S^1\times D^{n-1})$ is given by

$$\begin{align*}D^2\times D^{n-2}\cong(D^2\times D^{n-2}) \cup_{S^1\times D^{n-2}}(S^1\times D^{n-1}_+)\hookrightarrow (D^2\times S^{n-2})\cup_{S^1\times S^{n-2}} (S^1\times D^{n-1}), \end{align*}$$

where $D^{n-1}_+\subseteq D^{n-1}$ denotes the upper half-ball. We use the obvious embedding on each factor and embed $D^{n-2}\subseteq S^{n-2}$ as the upper half-sphere.

Hence, if $\varphi $ is trivial, then it is isotopic to $\varphi _0$ (as noted after Definition 3.7), so the manifold obtained by surgery along $\varphi $ is diffeomorphic to

(4.1) $$ \begin{align} M\# (D^2\times S^{n-2}\cup_{\mathrm{id}_{S^1\times S^{n-2}}} D^2\times S^{n-2})\cong M\# (S^2\times S^{n-2}). \end{align} $$

If $\varphi $ is nontrivial, then $\varphi \circ \tilde {\alpha }$ is trivial, where $\tilde {\alpha }\colon S^1\times D^{n-1}\to S^1\times D^{n-1}$ is defined by $\tilde {\alpha }(\lambda ,v)=(\lambda ,\alpha _\lambda v)$ and $\alpha $ is a smooth representative of the unique nontrivial class in $\pi _1(\mathrm {SO}(n-1))$ . Hence, $\varphi $ is isotopic to $\varphi _0\circ \tilde {\alpha }$ , so the manifold obtained by surgery along $\varphi $ is diffeomorphic to

(4.2) $$ \begin{align} M\# (D^2\times S^{n-2}\cup_{\tilde{\alpha}} D^2\times S^{n-2})\cong M\# (S^2\mathrel{\widetilde{\times} } S^{n-2}). \end{align} $$

Proof. By Proposition 3.6, since $w_2(\widetilde {M})\neq 0$ , the two nonequivalent normal framings of an embedded null-homotopic circle $f\colon S^1\hookrightarrow M$ are isotopic, so surgery along these framings results in diffeomorphic manifolds.

Now, fix an embedded 2-disc bounded by f together with a normal framing that extends over this 2-disc. Then a normal framing representing the other equivalence class does not extend over this disc. By Lemma 4.1, if we perform surgery along these normal framings, we therefore obtain the manifolds $M\#(S^2\times S^{n-2})$ and $M\#(S^2\mathrel {\widetilde {\times } } S^{n-2})$ , respectively.

Proof. We give the homotopy explicitly as follows. Define

$$\begin{align*}\Psi\colon[0,1]\times (W\cup_N [0,1]\times N)\to M\end{align*}$$

by $\Psi _t(p)=\Phi _t(\iota (p))$ for $p\in W$ and $\Psi _t(s,p)=\Phi _{(1-s)t+s}(\iota (p))$ for $(s,p)\in [0,1]\times N$ . Then $\Psi _0=\iota _\phi $ , and $\Psi _1$ equals $\Phi _1\circ \iota $ on W and $\Phi _1\circ \iota \circ \mathrm {pr}_N$ on $[0,1]\times N$ , which, under the identification $W\cong W\cup _N([0,1]\times N)$ is homotopic rel N to $\Phi _1\circ \iota $ . Further, $\Psi _t|_N=\Phi _1\circ \iota |_N$ for all $t\in [0,1]$ , showing that $\Psi $ is a homotopy rel N.

Proof of Proposition 4.3.

The long exact sequence in homology for the pair $(M,M\setminus \varphi (S^1\times D^{n-1}))$ yields the exact sequence

(4.3) $$ \begin{align} H_3(M,M\setminus \varphi(S^1\times D^{n-1}))\to H_2(M\setminus \varphi(S^1\times D^{n-1}))\to H_2(M)\to H_2(M,M\setminus \varphi(S^1\times D^{n-1})). \end{align} $$

By excision,

$$\begin{align*}H_i(M,M\setminus \varphi(S^1\times D^{n-1}))\cong H_i(S^1\times D^{n-1},S^1\times S^{n-2})=0\end{align*}$$

for $i=2,3$ , as $n\geq 5$ . Hence, the map $H_2(M\setminus \varphi (S^1\times D^{n-1}))\to H_2(M)$ is an isomorphism, that is,

(4.4) $$ \begin{align} H_2(M\setminus \varphi(S^1\times D^{n-1}))\cong H_2(M). \end{align} $$

Now, consider the manifold $M_i$ for $i=0,1$ . The long exact sequence in homology for the pair $(M_i,M_i\setminus (D^2\times S^{n-2}))$ yields the exact sequence

(4.5) $$ \begin{align} &H_3(M_i,M_i\setminus (D^2\times S^{n-2})) \longrightarrow H_2(M_i\setminus (D^2\times S^{n-2})) \longrightarrow H_2(M_i)\\\longrightarrow & H_2(M_i,M_i\setminus (D^2\times S^{n-2}))\longrightarrow H_1(M_i\setminus (D^2\times S^{n-2}))\longrightarrow H_1(M_i). \nonumber \end{align} $$

As for the pair $(M,M\setminus f(S^1\times D^{n-1}))$ , by excision,

$$\begin{align*}H_j(M_i,M_i\setminus (D^2\times S^{n-2}))\cong H_j(D^2\times S^{n-2},S^1\times S^{n-2})\end{align*}$$

and $H_j(D^2\times S^{n-2},S^1\times S^{n-2})$ vanishes for $j=3$ and is isomorphic to $\mathbb {Z}$ if $j=2$ . Further, since the inclusion $M_i\setminus (D^2\times S^{n-2})\hookrightarrow M_i$ induces an isomorphism on fundamental groups, we may rewrite the exact sequence (4.5) as

(4.6) $$ \begin{align} 0\longrightarrow H_2(M_i\setminus (D^2\times S^{n-2}))\longrightarrow H_2(M_i)\longrightarrow H_2(D^2\times S^{n-2},S^1\times S^{n-2})\longrightarrow0. \end{align} $$

By construction (cf. equations (4.2) and (4.2)), the element $x_i\in H_2(M_i)$ maps to a generator of $H_2(D^2\times S^{n-2},S^1\times S^{n-2})$ in equation (4.6) since a generator of the latter is represented by $D^2\times \{v\}$ for any $v\in S^{n-2}$ . We choose v (by possibly modifying the map $\alpha $ ) so that $\alpha _\lambda (v)=v$ for all $\lambda \in S^1$ and we denote by S the embedded $2$ -disc in the first $(D^2\times S^{n-2})$ -factor in equation (4.2) or equation (4.2) given by $D^2\times \{v\}$ . Hence, when glued to $D^2\times \{v\}$ in the second $(D^2\times S^{n-2})$ -factor in equation (4.2) or equation (4.2), the disc S represents the class $x_i$ . Note that in $M\setminus \varphi (S^1\times D^{n-1})$ , the surface S has boundary $\varphi (S^1\times \{v\})$ .

Since $\Phi _0=\text {id}_M$ , the diffeomorphism $\Phi _1$ is isotopic to a map that fixes $M\setminus \varphi (S^1\times D^{n-1})$ pointwise. Hence, $\Phi _1$ induces the identity in homology for all classes in $H_*(M)$ . Now, by Lemma 4.4, the inclusion of S, which we will denote by $\iota _S$ , followed by $\Phi _1$ is homotopic rel $\varphi (S^1\times \{v\})$ to $\iota _S$ extended by the map $\tilde {F}\colon [0,1]\times S^1\to M$ defined by $\tilde {F}(t,\lambda )=\Phi _t(\varphi (\lambda ,v))$ , where we identify S with $S\cup _{S^1}([0,1]\times S^1)$ . It follows that the surface representing $x_0$ is mapped to a surface which represents the class $x_1+\xi $ . This can be seen in a similar way as in the proof of Lemma 3.5: The map $\tilde {F}$ and the inclusions of S and $(D^2\times \{v\})$ all coincide on their boundaries, hence they define a map from $T^2$ with $2$ discs glued into $S^1\times \{1\}$ , and therefore define a map from the twice punctured solid torus into $M_1$ . The map restricted to each boundary component represents $\xi $ , $x_1$ and $\Phi _* x_0$ , respectively. Hence, after a suitable choice of orientations, we obtain $\Phi _* x_0=x_1+\xi $ .

Finally, since $H_1(M_i)\cong H_1(M)$ , the statement on the cohomology follows from the universal coefficient theorem.

Proof. We first consider the case where $M_1$ and $M_2$ are both spin, that is, they are both diffeomorphic to $\#_k (S^2\times S^{n-2})$ , which we will denote by $N_k$ . Denote a generator of $H^2(S^2\times S^{n-2})$ in the i-th summand of $N_k$ by $x_i$ . Then $(x_1,\dots ,x_k)$ is a basis of $H^2(N_k)$ . The automorphism group of $H^2(N_k)$ can be identified with $\mathrm {GL}(k,\mathbb {Z})$ and, by applying Corollary 4.5 to the i-th summand of $N_k$ with $\xi $ a multiple of the dual of $x_j$ , $i\neq j$ , we obtain that all elementary matrices are induced by a diffeomorphism. Since the elementary matrices together with the permutation matrices, which are obviously induced by diffeomorphisms, generate $\mathrm {GL}(k,\mathbb {Z})$ , the claim follows.

If $M_1$ and $M_2$ are nonspin, by applying Corollary 4.2 (possibly multiple times), we can assume that both $M_1$ and $M_2$ are diffeomorphic to a fixed connected sum of copies of $S^2\times S^{n-2}$ and $S^2\mathrel {\widetilde {\times } } S^{n-2}$ , where the latter appears at least once, and we denote this manifold by $N_k'$ . As before, we denote by $x_i$ a generator of the second cohomology of the i-th summand of $N_k'$ , so $(x_1,\dots ,x_k)$ is a basis of $H^2(M_k')$ . By Corollaries 4.5 and 4.6, we see as in the spin case that every automorphism of $H^2(N_k')$ is induced by a diffeomorphism if we allow the bundle structure of the summands to change. By restricting to those automorphisms that fix $w_2(N_k')$ we obtain all diffeomorphisms that do not change the bundle structures of the summands, that is, all self-diffeomorphisms of $N_k'$ .

Proof of Theorem A.

We will follow the same strategy as in [Reference Duan10, Theorem B]. We write $P_1$ as

$$\begin{align*}P_1\cong P_1\# S^{n+1}\cong P_1\# (D^2\times S^{n-1}\cup_{S^1\times S^{n-1}} S^1\times D^n).\end{align*}$$

Now the inclusion $\varphi $ of $S^1\times D^n$ into the second factor is a trivial normal framing as in the proof of Lemma 4.1. By Proposition 3.10, the normal framings $\varphi _{\pi _1}$ and $\varphi $ are isotopic if and only if $\widetilde {B}_1$ is not spin, and if $\widetilde {B}_1$ is spin, then $\varphi _{\pi _1}$ is isotopic to $\varphi \circ \tilde {\alpha }$ , where we extend $\tilde {\alpha }$ to $S^1\times D^n$ in the obvious way. It follows that

$$ \begin{align*} P&\cong (P_1\setminus \varphi_{\pi_1}(S^1\times D^n)^\circ)\cup_{S^1\times S^{n-1}} (P_2\setminus \varphi_{\pi_2}(S^1\times D^n)^\circ)\\ &\cong P_1\#(D^2\times S^{n-1})\cup_{S^1\times S^{n-1}} (P_2\setminus \varphi_{\pi_2}(S^1\times D^n)^\circ), \end{align*} $$

and we use either $\mathrm {id}_{S^1\times S^{n-1}}$ or $\tilde {\alpha }$ as gluing map, depending on whether $\varphi _\pi $ is trivial or not. In the first case, we obtain $P\cong P_1\# \Sigma _{e_2} M_2$ and in the second case, we obtain $P\cong P_1\# \widetilde {\Sigma }_{e_2} M_2$ .

Proof. As before, we denote by $P\xrightarrow {\pi }M$ the principal $S^1$ -bundle over M with Euler class e. The spaces $\Sigma _e M$ and $\widetilde {\Sigma }_e M$ fit into the following pushout diagram:

Hence, by van Kampen’s theorem, both $\pi _1(\Sigma _e M)$ and $\pi _1(\widetilde {\Sigma }_e M)$ are isomorphic to the quotient of $\pi _1(P\setminus (\varphi _\pi (D^n\times S^1)^\circ ))$ by the subgroup generated by the class represented by a fiber. By the long exact sequence of homotopy groups for the $S^1$ -bundle $P\setminus (\varphi _\pi (D^n\times S^1)^\circ )\xrightarrow {\pi } M\setminus {D^n}^\circ $ , this quotient is isomorphic to $\pi _1(M\setminus {D^n}^\circ )$ , which is isomorphic to $\pi _1(M)$ if $n\geq 3$ . This proves item (1) and item (2) follows from the Mayer–Vietoris sequence for the same pushout diagram.

Now, assume that M is simply connected. We consider $M'=(S^2\times S^{n-2})\# M$ , and denote by $P'\xrightarrow {\pi '}M'$ the principal $S^1$ -bundle with Euler class $e'=x^*+e$ , where $x^*$ denotes a generator of $H^2(S^2\times S^{n-2})$ . By Theorem A, it follows that

$$\begin{align*}P'\cong (S^3\times S^{n-2})\# \widetilde{\Sigma}_e M.\end{align*}$$

By the Gysin sequence, we have the following exact sequence:

$$\begin{align*}H^0(M')\xrightarrow{\cdot\smile e' } H^2(M')\xrightarrow{\pi^\prime*}H^2(P')\to 0. \end{align*}$$

Hence,

By Lemma 2.5, $w_2(P')=\pi ^\prime *w_2(M')=\pi ^\prime *w_2(M)$ , which only lies in $\langle e'\ \mod 2\rangle $ when $w_2(M)$ is trivial.

For $\Sigma _e M$ , we proceed similarly by defining $M'=(S^2\mathrel {\widetilde {\times } } S^{n-2})\# M$ . In this case, since $w_2(S^2\mathrel {\widetilde {\times } } S^{n-2})$ is nontrivial, $w_2(M')$ lies in $\langle e'\ \mod 2\rangle $ if and only if $w_2(M)\equiv e\ \mod 2$ . This proves item (3).

Proof. Recall that we have a local trivialization

$$\begin{align*}\varphi_\xi\colon D^n\times S^m\hookrightarrow E. \end{align*}$$

Thus, after smoothing corners, the restriction of $\varphi _\xi $ to $D^n\times S^m_+\cong D^n\times D^m$ , where $S^m_+$ denotes the (closed) upper hemisphere of $S^m$ , is an orientation-preserving embedding of $D^{m+n}$ into E.

It follows that in the decomposition

$$\begin{align*}\pi^*(E)\cong \pi^*(E\setminus\varphi_\xi(D^n\times S^m))\cup_{\text{id}_{S^{n-1}\times S^m\times S^1}}(D^n\times S^m\times S^1) \end{align*}$$

a local trivialization for $\pi ^*(E)$ is given by the inclusion of $D^n\times S^m_+\times S^1$ into the right-hand side. Hence, to construct the space $\Sigma _{\xi ^*c}E$ (resp. $\widetilde {\Sigma }_{\xi ^*c}E$ ), we need to glue the product $S^{n+m-1}\times D^2$ to $\pi ^*(E\setminus \varphi _\xi (D^n\times S^m))\cup _{\text {id}_{S^{n-1}\times S^m_-\times S^1}}(D^n\times S^m_-\times S^1)$ along the boundary $S^{n+m-1}\times S^1$ , which, in this decomposition, is given by

$$\begin{align*}(S^{n-1}\times S^m_+\times S^1)\cup_{\text{id}_{S^{n-1}\times S^{m-1}\times S^1}}(D^{n}\times S^{m-1}\times S^1). \end{align*}$$

If we now decompose

$$\begin{align*}(S^{n+m-1}\times D^2)\cong (S^{n-1}\times S^m_+\times D^2)\cup_{\text{id}_{S^{n-1}\times S^{m-1}\times D^2}}(D^{n}\times S^{m-1}\times D^2), \end{align*}$$

we obtain that the space $\Sigma _{\xi ^*c}E$ (resp. $\widetilde {\Sigma }_{\xi ^*c}E$ ) is the result of gluing according to the following diagram, where the map $\phi $ will be constructed below:

(5.1)

Here, an arrow denotes gluing of the two spaces it connects along parts of their boundary via the map indicated.

The map $\phi $ in diagram (5.1) is a self-diffeomorphism of

$$\begin{align*}(S^{n-1}\times S^m_+\times S^1)\cup_{\mathrm{id}_{S^{n-1}\times S^{m-1}\times S^1}}(D^n\times S^{m-1}\times S^1)\cong S^{n+m-1}\times S^1 \end{align*}$$

defined as follows: For $\Sigma _{\xi ^*c}E$ , set $\phi =\mathrm {id}_{S^{n+m-1}\times S^1}$ . For $\widetilde {\Sigma }_{\xi ^*c}E$ , let $\alpha $ be a smooth representative of a generator of $\pi _1(\mathrm {SO}(n))$ (which is isomorphic to $\mathbb {Z}/2$ if $n>2$ and to $\mathbb {Z}$ if $n=2$ ), and set

$$\begin{align*}\phi(x,y,\lambda)=(T_\lambda x,y,\lambda). \end{align*}$$

We claim that $\phi $ is the gluing map in the construction of $\Sigma _{\xi ^*c}E$ (resp. $\widetilde {\Sigma }_{\xi ^*c}E$ ). For $\Sigma _{\xi ^*c}E$ , this is clear by construction. For $\widetilde {\Sigma }_{\xi ^*c}E$ , note that in the decomposition

$$\begin{align*}S^{n+m-1}\cong (S^{n-1}\times S^m_+)\cup_{\mathrm{id}_{S^{n-1}\times S^{m-1}}}(D^n\times S^{m-1}) \end{align*}$$

the first factor corresponds to the embedding of a tubular neighborhood of $S^{n-1}\subseteq \mathbb {R}^n\subseteq \mathbb {R}^{n+m}$ into $S^{n+m-1}\subseteq \mathbb {R}^{n+m-1}$ . Since the inclusion $\mathrm {SO}(n)\subseteq \mathrm {SO}(n+m)$ induces a surjection on fundamental groups (and in fact an isomorphism if $n>2$ ), it follows that the map $\phi $ represents the nontrivial class in $\pi _1(\mathrm {SO}(n+m))$ .

We now modify diagram (5.1) by noting that the map $\phi |_{S^n\times S^{m-1}\times S^1}$ extends over $D^n\times S^m_-\times S^1$ as the identity on the second factor, and we denote the extension again by $\phi $ . Hence, we obtain the following gluing diagram:

(5.2)

We observe now that gluing according to the right vertical part of diagram (5.2) yields the space

$$\begin{align*}(D^n\times S^m_-\times S^1)\cup_{\mathrm{id}_{D^n\times S^{m-1}\times S^1}}(D^n\times S^{m-1}\times D^2)\cong (D^n\times S^{m+1}), \end{align*}$$

while gluing according to the left vertical part yields the space

$$ \begin{align*} \pi^*(E\setminus\varphi_\xi(D^n\times S^m))\cup_{\phi|_{S^{n-1}\times S^m_+\times S^1}}(S^{n-1}\times D^m\times D^2), \end{align*} $$

which can be alternatively written as

$$ \begin{align*} \pi^*(E\setminus\varphi_\xi(D^n\times S^m))\cup_{\phi|_{S^{n-1}\times S^m_+\times S^1}}((S^{n-1}\times S^m\times D^2)\setminus (S^{n-1}\times S^m_-\times D^2)). \end{align*} $$

This space is, by construction, the space $E(\Sigma _e\xi )$ (resp. $E(\widetilde {\Sigma }_e\xi )$ ) with the image of the embedding $\iota _\xi $ (resp. $\widetilde {\iota }_\xi $ ) removed. It follows that $\Sigma _{\xi ^*e}E$ (resp. $\widetilde {\Sigma }_{\xi ^*e}E$ ) is obtained from $E(\Sigma _e\xi )$ (resp. $E(\widetilde {\Sigma }_e\xi )$ ) by surgery along $\iota _\xi $ (resp. $\widetilde {\iota }_\xi $ ).

Proof. (1). By definition, we can decompose the spaces $E(\Sigma _0 \xi )$ and $E(\widetilde {\Sigma }_0\xi )$ as

(5.3) $$ \begin{align} E(\Sigma_0\xi)\cong (D^n\times S^1\times S^m)\cup_{\phi_1}(S^{n-1}\times D^2\times S^m) \end{align} $$

and

(5.4) $$ \begin{align} E(\widetilde{\Sigma}_0\xi)\cong (D^n\times S^1\times S^m)\cup_{\phi_2}(S^{n-1}\times D^2\times S^m), \end{align} $$

where the diffeomorphisms $\phi _1,\phi _2\colon S^{n-1}\times S^1\times S^m\to S^{n-1}\times S^1\times S^m$ are given by

$$\begin{align*}\phi_1(x,y,z)=(x,y,T_x z) \end{align*}$$

and

$$\begin{align*}\phi_2(x,y,z)=(\alpha_y x,y, T_x z). \end{align*}$$

We further decompose

$$\begin{align*}S^{n-1}\times D^2\times S^m\cong (S^{n-1}\times D^2\times S^m_+)\cup_{\mathrm{id}_{S^{n-1}\times D^2\times S^{m-1}}}(S^{n-1}\times D^2\times S^m_-), \end{align*}$$

and the embeddings $\iota _\xi $ and $\widetilde {\iota }_\xi $ are given by the inclusion of the second factor.

Since the image of T is contained in $\mathrm {SO}(m)$ , we can assume that $T_x$ preserves $S^m_-$ and is given by a linear map on $S^m_-$ when identifying $S^m_-\cong D^m$ . In particular, $T_x$ fixes the south pole $z_S\in S^m_-$ . Further, we can deform the map $\alpha $ to be constant $\mathrm {id}_{\mathbb {R}^n}$ on $S^1_-$ .

By isotoping the embeddings $\iota _\xi $ and $\widetilde {\iota }_\xi $ to the left-hand side of equations (5.3) and (5.4), respectively, we obtain in both cases the embedding

$$\begin{align*}\iota\colon S^{n-1}\times D^1\times S^1_-\times S^m_-\hookrightarrow D^n\times S^1\times S^m,\quad (x,y_1,y_2,z)\mapsto ((x,\frac{1}{2}y_1),y_2,T_x^{-1}z), \end{align*}$$

where we have identified $D^2\cong D^1\times S^1_-$ and $D^n$ as the space obtained from $S^{n-1}\times D^1=S^{n-1}\times [-1,1]$ by collapsing $S^{n-1}\times \{-1\}$ to a point.

Now, define the map $T^\prime \colon S^{n-1}\to \mathrm {SO}(m+2)$ ,

$$\begin{align*}T^{\prime}_x(y,z)=T_xz \end{align*}$$

for $x\in S^{n-1}$ , $y\in \mathbb {R}^2$ and $z\in \mathbb {R}^m$ . Then, when viewing $\iota $ as a normal framing of an embedding of $S^{n-1}$ , modifying the framing by $T'$ yields a normal framing that extends over an embedded disc. It follows as in Lemma 4.1 (see, e.g., [Reference Reiser42, Lemma 3.8]) that the manifold obtained by surgery along the embedding $\iota _\xi $ , which by Proposition 5.5 is diffeomorphic to $\Sigma _0 E$ , is diffeomorphic to

$$\begin{align*}E(\Sigma_0\xi)\#((D^n\times S^{m+1})\cup_{\tilde{T}'}(D^n\times S^{m+1}), \end{align*}$$

where $\tilde {T}'\colon S^{n-1}\times S^{m+1}\to S^{n-1}\times S^{m+1}$ is defined by $\tilde {T}'(x,y)=(x,T^{\prime }_x y)$ , and similarly for $E(\widetilde {\Sigma }_0\xi )$ . The right-hand side is the total space of the linear $S^{m+1}$ -bundle over $S^n$ with clutching function $T'$ .

(2). By Proposition 5.5, the manifold $\Sigma _{\xi ^* e}E$ (resp. $\widetilde {\Sigma }_{\xi ^* e}E$ ) is obtained by surgery on an embedding of $S^1\times D^{m+2}$ in $E(\Sigma _e\xi )$ (resp. $E(\widetilde {\Sigma }_e\xi )$ ). The spaces $E(\Sigma _e\xi )$ and $E(\widetilde {\Sigma }_e\xi )$ are total spaces of linear sphere bundles over $\Sigma _e S^2$ and $\widetilde {\Sigma }_e S^2$ , respectively, which, by Example 5.3, are diffeomorphic to $S^3$ . Since any linear sphere bundle over $S^3$ is trivial, both $E(\Sigma _e\xi )$ and $E(\widetilde {\Sigma }_e\xi )$ are diffeomorphic to $S^3\times S^m$ .

Since $S^3\times S^m$ is simply connected, it follows from Lemma 4.1 that $\Sigma _{\xi ^* e}E$ (resp. $\widetilde {\Sigma }_{\xi ^* e}E$ ) is diffeomorphic to either $(S^2\times S^{m+1})\# (S^3\times S^m)$ (which is spin) or $(S^2\mathrel {\widetilde {\times } } S^{m+1})\# (S^3\times S^m)$ (which is nonspin). By Lemma 5.2, we can characterize when $\Sigma _{\xi ^* e}E$ (resp. $\widetilde {\Sigma }_{\xi ^* e}E$ ) in terms of the Euler class and Stiefel–Whitney class of E, which yields the different cases as claimed.

Proof of Theorem B.

Item (1) follows from Proposition 3.10 and Lemma 4.1 and items (2) and (3) follow from Proposition 5.6.

Proof. Since

$$\begin{align*}(S^2\mathrel{\widetilde{\times} } S^{n-2})\# (S^2\times S^{n-2})\cong (S^2\mathrel{\widetilde{\times} } S^{n-2})\# (S^2\mathrel{\widetilde{\times} } S^{n-2}) \end{align*}$$

by Corollary 4.2, the information whether M is spin or not is sufficient (together with the Betti numbers) to determine its diffeomorphism type. All other claims are obvious.

Proof of Theorem 6.1.

Let $\xi _0\in H_2(B_2)$ be a class such that $e_2\frown \xi _0=d_2$ . We first assume that $b_2(B_1)\geq 2$ . Let $k,l\in \mathbb {Z}$ so that $k d_1+l d_2=1$ (which exist since e is primitive). Denote by $x_i^*$ a generator of the second cohomology of the i-th summand in $B_1$ that is a sphere bundle over $S^2$ . Then, by Corollary 4.7, we can apply a self-diffeomorphism of $B_1$ so that $e_1$ is given by $k d_1 x_1^*+d_1 x_2^*$ . Hence, if we write $B_1\# B_2$ as $M\# N$ , where N is the summand of $B_1$ with $H^2(N)$ generated by $x_1^*$ and M is the connected sum of all remaining summands, we can apply Corollary 4.5 or 4.6 with $\xi =l\xi _0$ (and note that the class $l\xi _0$ can be represented by a map $S^2\to B_2$ by the Hurewicz theorem). Thus, we obtain a self-diffeomorphism of $B_1\# B_2$ that maps e to $x_1^*+d_1 x_2^*+e_2$ . Hence, the restriction of e to one $(S^2\times S^{n-2})$ or $(S^2\mathrel {\widetilde {\times } } S^{n-2})$ -summand is primitive.

In case $b_2(B_1)=1$ , we obtain the same conclusion by applying Corollary 4.5 or 4.6 to $\xi =l\xi _0$ , where here $l\in \mathbb {Z}$ is chosen so that $d_1+l d_2=\pm 1$ .

We now repeatedly apply Theorem A to obtain that P is the connected sum of $(S^3\times S^{n-2})$ (which is the total space of the principal $S^1$ -bundle over $(S^2\times S^{n-2})$ or $(S^2\mathrel {\widetilde {\times } } S^{n-2})$ with Euler class $x_1^*$ ) and twisted suspensions of $B_2$ along $e_2$ and of the remaining products of spheres or $(S^2\mathrel {\widetilde {\times } } S^{n-2})$ -summands. Thus, the claim now follows from Theorem B.

Proof of Corollary D.

We first consider the case $k=1$ and assume that M is of the form (*) and its partial Euler characteristics satisfy the stated conditions. We define the manifold B such that B is of the form (*), has dimension $n-1$ , is nonspin and has Betti numbers $b_i(B)=(-1)^i\chi _i(M)$ for all $i=2,\dots ,\lfloor \frac {n-1}{2}\rfloor $ . Note that, if $n-1$ is even, then $b_{\frac {n-1}{2}}(B)$ is even by assumption, so B is well defined and unique by Lemma 6.2.

Now, let $e\in H^2(B)$ be a primitive element that satisfies $e\equiv w_2(B)\ \mod 2$ if and only if M is spin. We define P as the total space of the principal $S^1$ -bundle over B with Euler class e. By Theorem C, the manifold P is of the form (*) and satisfies the following conditions:

• • $b_2(P)=b_2(B)-1=\chi _2(M)-1=b_2(M)$ .

• • For $2<i<\lfloor \frac {n-1}{2}\rfloor $ , or $i=\frac {n-1}{2}$ if $n-1$ is even, we have

  $$\begin{align*}b_i(P)=b_{i-1}(B)+b_i(B)=(-1)^{i-1}\chi_{i-1}(M)+(-1)^i\chi_i(M)=b_i(M).\end{align*}$$

• • If n is even, we have

  $$\begin{align*}b_{\frac{n}{2}}(P)=2b_{\frac{n-2}{2}}(B)=2(-1)^{\frac{n}{2}-1}\chi_{\frac{n}{2}-1}(M)=b_{\frac{n}{2}}(M),\end{align*}$$
  since $0=\chi (M)=2\chi _{\frac {n}{2}-1}(M)+(-1)^{\frac {n}{2}}b_{\frac {n}{2}}(M)$ .

Since P is spin if and only if M is spin by Lemma 2.3, it follows that P is diffeomorphic to M by Lemma 6.2. Hence, M admits a free $S^1$ -action with quotient of the form (*).

For general k, we iterate the above argument to obtain a sequence $M\cong P_k\xrightarrow {\pi _k}\dots \xrightarrow {\pi _1} P_0$ of principal $S^1$ -bundles with Euler classes $e(\pi _i)\in H^2(P_{i-1})$ , so that each $P_i$ is of the form (*). Let $e_i\in H^2(P_0)$ so that $e(\pi _i)=\pi _{i-1}^*\dots \pi _1^* e_i$ , which exists since each $e(\pi _i)$ is primitive and the induced map on $H^2$ of each $\pi _i$ can be identified with the quotient map by $e_i$ , by Lemma 2.3. Define P as the principal $T^k$ -bundle with Euler class $(e_1,\dots ,e_k)$ . Then, by Lemma 2.2, P is diffeomorphic to $P_k\cong M$ , showing that M admits a free $T^k$ -action with quotient $P_0$ , which is of the form (*).

Conversely, assume that M admits a free $S^1$ -action with quotient of the form (*). Then M is the total space of a principal $S^1$ -bundle with base B of the form (*) and it follows inductively from Theorem C that $b_i(B)=(-1)^i\chi _i(M)$ for $i=2,\dots ,\lfloor \frac {n-1}{2}\rfloor $ , which is nonnegative since $b_i(B)\geq 0$ . Further, by Lemma 6.2, we have that $(-1)^{\frac {n-1}{2}}\chi _{\frac {n-1}{2}}(M)$ is even when $n-1$ is even, and, since $b_{\frac {n}{2}}(M)=2b_{\frac {n}{2}-1}(B)=(-1)^{\frac {n}{2}-1}\chi _{\frac {n}{2}-1}(M)$ if n is even, it also follows that $\chi _n(M)=0$ if n is even. The statement for general k now follows by induction.

Proof. By definition, the total space of the sphere bundle $S(T)\to P(E)$ of the tautological line bundle $T\to P(E)$ is given by

$$\begin{align*}S(T)=\{(v,\varphi)\in E\times P(E)\mid v\in\varphi,\, \lVert v\rVert=1\}. \end{align*}$$

By projection onto the first coordinate, we obtain an identification of $S(T)$ with the total space $S(E)$ of the sphere bundle of $\xi $ . Since $w_2(\xi )=c_1(\xi )\ \mod 2$ , the claim follows.

Proof. We define $\overline {E}\xrightarrow {\pi } \mathbb {C} P^m$ as the sum of the tautological line bundle with the trivial bundle $\underline {\mathbb {R}}^{r-1}_{\mathbb {C} P^m}$ . Then $w_2(\pi )$ is nontrivial (see, e.g., [Reference Milnor and Stasheff32, Theorem 14.4]). If $E\to \mathbb {C} P^m$ denotes the corresponding sphere bundle, we have $TE\oplus \underline {\mathbb {R}}_E\cong \pi ^*T\mathbb {C} P^m\oplus \pi ^*\overline {E}$ , cf. Lemma 2.11. Hence, $w_2(E)$ is trivial if and only if $w_2(\mathbb {C} P^m)$ is nontrivial, which is the case if and only if m is even.

By construction, the bundle $P\to E$ fits into the following pull-back diagram:

Here, $S^{2m+1}\to \mathbb {C} P^m$ denotes the Hopf fibration (i.e., the principal $S^1$ -bundle whose Euler class is a generator of $H^2(\mathbb {C} P^m)$ ). It follows that $P\to S^{2m+1}$ is a linear $S^r$ -bundle. Since the structure group of this bundle is contained in $\mathrm {SO}(2)\cong S^1$ and since $S^1$ has trivial higher homotopy groups, this bundle is trivial, so $P\cong S^{2m+1}\times S^r$ .

Proof of Theorem E.

Let $E(m,r)$ denote the total space of the linear $S^r$ -bundle over $\mathbb {C} P^m$ from Lemma 6.5. We then set

$$\begin{align*}E_m^r=\begin{cases} \mathbb{C} P^m\times S^r,\quad & m\text{ odd},\\ E(m,r),\quad & m\text{ even}, \end{cases}\quad \text{ and }\widetilde{E}_m^r=\begin{cases} E(m,r),\quad & m\text{ even},\\ \mathbb{C} P^m\times S^r,\quad & m\text{ odd} \end{cases} \end{align*}$$

so that $E_m^r$ is spin and $\widetilde {E}_m^r$ is nonspin, and the principal $S^1$ -bundle over $E_m^r$ or $\widetilde {E}_m^r$ whose Euler class is the pull-back of a generator of $H^2(\mathbb {C} P^m)$ has total space $S^{2m+1}\times S^r$ .

(1). We define

$$\begin{align*}B=\begin{cases} \left(\#_l\mathbb{C} P^{\frac{n-1}{2}}\right)\#\left(\#_{i=1}^{\frac{n-3}{2}}\#_{b_{2i+1}(M)}E_i^{n-2i-1}\right),\quad & n\equiv 3\quad \mod 4,\\ \left(\#_l\mathbb{C} P^{\frac{n-1}{2}}\right)\#\left(\#_{i=1}^{\frac{n-3}{2}}\#_{b_{2i+1}(M)}\widetilde{E}_i^{n-2i-1}\right),\quad & n\equiv 1\quad \mod 4\\ \end{cases} \end{align*}$$

for some $l\geq 0$ . Let $e\in H^2(B)$ be a class that restricts to a generator of $H^2(\mathbb {C} P^{\frac {n-1}{2}})$ on each $\mathbb {C} P^{\frac {n-1}{2}}$ -summand and to the pull-back of a generator of $H^2(\mathbb {C} P^i)$ on each $E_i^j$ and $\widetilde {E}_i^j$ -summand. Then, for the principal $S^1$ -bundle $P\to B$ with Euler class e, we have by Theorems A and B and Lemma 6.5 (note that the summands of B are either all spin or all nonspin)

$$\begin{align*}P\cong \#_a (S^2\times S^{n-2})\#_{i=1}^{\frac{n-3}{2}}\#_{b_{2i+1(M)}}(S^{2i+1}\times S^{n-2i-1}), \end{align*}$$

where $a=l-1+\sum _{i=1}^{\frac {n-5}{2}} b_{2i+1}(M)$ . Thus, since $b_{2i+1}(M)= b_{n-2i-1}(M)$ , we have for $3\leq i\leq \frac {n-1}{2}$ , that $b_i(P)=b_i(M)$ . Hence, if M is spin, it becomes diffeomorphic to P after connected sum with sufficiently many copies of $(S^2\times S^{n-2})$ and by choosing l large enough.

For the nonspin case or the case where M is spin and we take connected sums with copies of $(S^2\mathrel {\widetilde {\times } } S^{n-2})$ , we replace one $E_i^j$ -summand in B by $\widetilde {E}_i^j$ or vice versa, provided there is a nontrivial summand of this form. Then P has a summand of the form $(S^2\mathrel {\widetilde {\times } } S^{n-2})$ , hence the claim follows for l large enough by Corollary 4.2. If there exists no such summand, we additionally introduce a summand for B given by $E_{\frac {n-3}{2}}^2\#\widetilde {E}_{\frac {n-3}{2}}^2$ , which results in an additional summand for P given by $\#_4(S^2\mathrel {\widetilde {\times } } S^{n-2}) $ .

(2). We consider each dimension separately. First, note that the result in dimension 5 was shown in [Reference Duan and Liang11, Corollary 2] by proving that any $5$ -manifold of the form (*) is the total space of a principal circle bundle over a closed, simply connected 4-manifold. The six-dimensional case follows directly from Corollary D (see also [Reference Corro and Galaz-García7, Corollary B] and [Reference Duan10, Theorem C]).

For dimensions $7$ – $10$ , we summarize in Table 1 how the base manifold B in each case is given. One then easily verifies, using Theorems A and B, that the total space of the principal circle bundle over B with suitable Euler class e is diffeomorphic to M. By $P(E)$ we denote the total space of a projective bundle of a vector bundle with odd first Chern class of appropriate dimension (cf. Lemma 6.4). The Euler class e will always be the pull-back of a generator of $H^2(\mathbb {C} P^i)$ on each summand of the form $E_i^j$ , the Euler class of the tautological line bundle over $P(E)$ and a generator of the second cohomology on each summand of the form $S^2\times S^{n-2}$ and $S^2\mathrel {\widetilde {\times } } S^{n-2}$ .