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I am interested in the difference between block bundle and fiber bundle.

Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.

A block diffeomorphism of $\Delta^p\times M$ is a diffeomorphism $\Delta^p\times M\to \Delta^p\times M$ which for each face $\sigma \subset \Delta^p$ restricts to a diffeomorphism of $\sigma\times M$.

A block chart for $E$ over a simplex $\sigma\in K$ is a homeomorphism $h_{\sigma}:p^{-1}(\sigma)\to \sigma\times M$ which for every face $\tau$ restricts to a homeomorphism $p^{-1}(\tau)\to \tau\times M$.

A block atlas is a set $\mathcal{A}$ of block charts, at least one over each simplex of K, such that if $h_{\sigma_i}:p^{-1}(\sigma_i)\to \sigma_i\times M$ for $i=0,1$ are two elements of $\mathcal{A}$ then the composition $h_{\sigma_1}\circ h_{\sigma_0}^{-1}$ from $(\sigma_0\cap\sigma_1)\times M$ to itself is a block diffeomorphism.

A block bundle structure is a maximal block atlas. The resulting structure is a block bundle.

This notion is very close to fiber bundle.

I am wondering if there exists a block bundle s.t. both fiber and base are manifolds but it does not admit fiber bundle structure.Is every $S^3$ block bundle over $S^4$ a fiber bundle?

(This may be reduced to a lifting problem,since the fiber bundle has classifying space $BO(4)$ and the concordance class of such block bundle has classifying space $B\widetilde{diff}(S^3)$.some knowledge about the homotopy group of $B\widetilde{diff}(S^3)$ and $\widetilde{diff}(S^3)/diff(S^3)$ would surely be helpful here.)

I am interested in the difference between block bundle and fiber bundle.

Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.

A block diffeomorphism of $\Delta^p\times M$ is a diffeomorphism $\Delta^p\times M\to \Delta^p\times M$ which for each face $\sigma \subset \Delta^p$ restricts to a diffeomorphism of $\sigma\times M$.

A block chart for $E$ over a simplex $\sigma\in K$ is a homeomorphism $h_{\sigma}:p^{-1}(\sigma)\to \sigma\times M$ which for every face $\tau$ restricts to a homeomorphism $p^{-1}(\tau)\to \tau\times M$.

A block atlas is a set $\mathcal{A}$ of block charts, at least one over each simplex of K, such that if $h_{\sigma_i}:p^{-1}(\sigma_i)\to \sigma_i\times M$ for $i=0,1$ are two elements of $\mathcal{A}$ then the composition $h_{\sigma_1}\circ h_{\sigma_0}^{-1}$ from $(\sigma_0\cap\sigma_1)\times M$ to itself is a block diffeomorphism.

A block bundle structure is a maximal block atlas. The resulting structure is a block bundle.

This notion is very close to fiber bundle.

I am wondering if there exists a block bundle s.t. both fiber and base are manifolds but it does not admit fiber bundle structure.Is every $S^3$ block bundle over $S^4$ a fiber bundle?

(This may be reduced to a lifting problem,since the fiber bundle has classifying space $BO(4)$ and the concordance class of such block bundle has classifying space $B\widetilde{Cat}(S^3)$.some knowledge about the homotopy group of $B\widetilde{Cat}(S^3)$ and $\widetilde{Cat}(S^3)/Cat(S^3)$ would surely be helpful here.$Cat=Diff,Top,PL$)

I am interested in the difference between block bundle and fiber bundle.

Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.

A block diffeomorphism of $\Delta^p\times M$ is a diffeomorphism $\Delta^p\times M\to \Delta^p\times M$ which for each face $\sigma \subset \Delta^p$ restricts to a diffeomorphism of $\sigma\times M$.

A block chart for $E$ over a simplex $\sigma\in K$ is a homeomorphism $h_{\sigma}:p^{-1}(\sigma)\to \sigma\times M$ which for every face $\tau$ restricts to a homeomorphism $p^{-1}(\tau)\to \tau\times M$.

A block atlas is a set $\mathcal{A}$ of block charts, at least one over each simplex of K, such that if $h_{\sigma_i}:p^{-1}(\sigma_i)\to \sigma_i\times M$ for $i=0,1$ are two elements of $\mathcal{A}$ then the composition $h_{\sigma_1}\circ h_{\sigma_0}^{-1}$ from $(\sigma_0\cap\sigma_1)\times M$ to itself is a block diffeomorphism.

A block bundle structure is a maximal block atlas. The resulting structure is a block bundle.

This notion is very close to fiber bundle.

I am wondering if there exist a block bundle s.t. both fiber and base are manifolds but it does not admit fiber bundle structure.Is every $S^3$ block bundle over $S^4$ a fiber bundle?

(This may be reduced to a lifting problem,since the fiber bundle has classifying space $BO(4)$ and the concordance class of such block bundle has classifying space $B\widetilde{diff}(S^3)$.some knowledge about the homotopy group of $B\widetilde{diff}(S^3)$ and $\widetilde{diff}(S^3)/diff(S^3)$ would surely be helpful here.)

I am interested in the difference between block bundle and fiber bundle.

Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.

A block diffeomorphism of $\Delta^p\times M$ is a diffeomorphism $\Delta^p\times M\to \Delta^p\times M$ which for each face $\sigma \subset \Delta^p$ restricts to a diffeomorphism of $\sigma\times M$.

A block chart for $E$ over a simplex $\sigma\in K$ is a homeomorphism $h_{\sigma}:p^{-1}(\sigma)\to \sigma\times M$ which for every face $\tau$ restricts to a homeomorphism $p^{-1}(\tau)\to \tau\times M$.

A block atlas is a set $\mathcal{A}$ of block charts, at least one over each simplex of K, such that if $h_{\sigma_i}:p^{-1}(\sigma_i)\to \sigma_i\times M$ for $i=0,1$ are two elements of $\mathcal{A}$ then the composition $h_{\sigma_1}\circ h_{\sigma_0}^{-1}$ from $(\sigma_0\cap\sigma_1)\times M$ to itself is a block diffeomorphism.

A block bundle structure is a maximal block atlas. The resulting structure is a block bundle.

This notion is very close to fiber bundle.

I am wondering if there exists a block bundle s.t. both fiber and base are manifolds but it does not admit fiber bundle structure.Is every $S^3$ block bundle over $S^4$ a fiber bundle?

(This may be reduced to a lifting problem,since the fiber bundle has classifying space $BO(4)$ and the concordance class of such block bundle has classifying space $B\widetilde{diff}(S^3)$.some knowledge about the homotopy group of $B\widetilde{diff}(S^3)$ and $\widetilde{diff}(S^3)/diff(S^3)$ would surely be helpful here.)

I am interested in the difference between block bundle and fiber bundle.

Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.

A block diffeomorphism of $\Delta^p\times M$ is a diffeomorphism $\Delta^p\times M\to \Delta^p\times M$ which for each face $\sigma \subset \Delta^p$ restricts to a diffeomorphism of $\sigma\times M$.

A block chart for $E$ over a simplex $\sigma\in K$ is a homeomorphism $h_{\sigma}:p^{-1}(\sigma)\to \sigma\times M$ which for every face $\tau$ restricts to a homeomorphism $p^{-1}(\tau)\to \tau\times M$.

A block atlas atlas is a set $\mathcal{A}$ of block charts, at least one over each simplex of K, such that if $h_{\sigma_i}:p^{-1}(\sigma_i)\to \sigma_i\times M$ for $i=0,1$ are two elements of $\mathcal{A}$ then the composition $h_{\sigma_1}\circ h_{\sigma_0}^{-1}$ from $(\sigma_0\cap\sigma_1)\times M$ to itself is a block diffeomorphism.

A block bundle structure is a maximal block atlas. The resulting structure is a block bundle.

This notion is very close to fiber bundle.

I am wondering if there exist a block bundle s.t. both fiber and base are manifolds but it does not admit fiber bundle structure.Is every $S^3$ block bundle over $S^4$ a fiber bundle?

(This may be reduced to a lifting problem,since the fiber bundle has classifying space $BO(4)$ and the concordance class of such block bundle has classifying space $B\widetilde{diff}(S^3)$.some knowledge about the homotopy group of $B\widetilde{diff}(S^3)$ and $\widetilde{diff}(S^3)/diff(S^3)$ would surely be helpful here.)

I am interested in the difference between block bundle and fiber bundle.

Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.

A block diffeomorphism of $\Delta^p\times M$ is a diffeomorphism $\Delta^p\times M\to \Delta^p\times M$ which for each face $\sigma \subset \Delta^p$ restricts to a diffeomorphism of $\sigma\times M$.

A block chart for $E$ over a simplex $\sigma\in K$ is a homeomorphism $h_{\sigma}:p^{-1}(\sigma)\to \sigma\times M$ which for every face $\tau$ restricts to a homeomorphism $p^{-1}(\tau)\to \tau\times M$.

A block atlas is a set $\mathcal{A}$ of block charts, at least one over each simplex of K, such that if $h_{\sigma_i}:p^{-1}(\sigma_i)\to \sigma_i\times M$ for $i=0,1$ are two elements of $\mathcal{A}$ then the composition $h_{\sigma_1}\circ h_{\sigma_0}^{-1}$ from $(\sigma_0\cap\sigma_1)\times M$ to itself is a block diffeomorphism.

A block bundle structure is a maximal block atlas. The resulting structure is a block bundle.

This notion is very close to fiber bundle.

I am wondering if there exist a block bundle s.t. both fiber and base are manifolds but it does not admit fiber bundle structure.Is every $S^3$ block bundle over $S^4$ a fiber bundle?

(This may be reduced to a lifting problem,since the fiber bundle has classifying space $BO(4)$ and the concordance class of such block bundle has classifying space $B\widetilde{diff}(S^3)$.some knowledge about the homotopy group of $B\widetilde{diff}(S^3)$ and $\widetilde{diff}(S^3)/diff(S^3)$ would surely be helpful here.)