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The answer is yes, see Rudyak's paper Piecewise linear structures on topological manifolds, Examples 21.4:

There are topological manifolds that can be triangulated as simplicial complexes but do not admit any PL structure.

Such examples exist in fact in any dimension $n \geq 5$, and are of the form $$M_k=V \times T^k, \quad k \geq 1,$$ where $V$ is the famous $E_8$-manifold constructed by Friedman. See Theorem 7.2 and Corollary 7.4 in the quoted paper.

It is worth remarking that the $4$-manifold $V$ is not triangulable as a simplicial complex. However, by the work of Siebenmann and others, it is known that every orientable topological 5-dimensional closed manifold can be triangulated as a simplicial complex, see Theorem 21.5. This implies that $M_1= V \times S^1$ is triangulable, so $M_k=M_1 \times T^{k-1}$ is triangulable for all $k \geq 1$.

The answer is yes, see Rudyak's paper Piecewise linear structures on topological manifolds, Examples 21.4:

There are topological manifolds that can be triangulated as simplicial complexes but do not admit any PL structure.

Such examples exist in fact in any dimension $n \geq 5$, and are of the form $$M_k=V \times T^k, \quad k \geq 1,$$ where $V$ is the famous $E_8$-manifold constructed by Freedman. See Theorem 7.2 and Corollary 7.4 in the quoted paper.

It is worth remarking that the $4$-manifold $V$ is not triangulable as a simplicial complex. However, by the work of Siebenmann and others, it is known that every orientable topological 5-dimensional closed manifold can be triangulated as a simplicial complex, see Theorem 21.5. This implies that $M_1= V \times S^1$ is triangulable, so $M_k=M_1 \times T^{k-1}$ is triangulable for all $k \geq 1$.

Yes, such examples do in fact exist. See Rudyak's paper Piecewise linear structures on topological manifolds, Examples 21.4:

There are topological manifolds that can be triangulated as simplicial complexes but do not admit any PL structure.

Such examples exist in fact in any dimension $n \geq 5$, and are of the form $$M_k=V \times T^k, \quad k \geq 1,$$ where $V$ is the famous $E_8$-manifold constructed by Friedman. See Theorem 7.2 and Corollary 7.4 in the quoted paper.

It is worth remarking that the $4$-manifold $V$ is not triangulable as a simplicial complex. However, by the work of Siebenmann and others, it is known that every orientable topological 5-dimensional closed manifold can be triangulated as a simplicial complex, see Theorem 21.5. This implies that $M_1= V \times S^1$ is triangulable, so $M_k=M_1 \times T^{k-1}$ is triangulable for all $k$.

The answer is yes, see Rudyak's paper Piecewise linear structures on topological manifolds, Examples 21.4:

There are topological manifolds that can be triangulated as simplicial complexes but do not admit any PL structure.

Such examples exist in fact in any dimension $n \geq 5$, and are of the form $$M_k=V \times T^k, \quad k \geq 1,$$ where $V$ is the famous $E_8$-manifold constructed by Friedman. See Theorem 7.2 and Corollary 7.4 in the quoted paper.

It is worth remarking that the $4$-manifold $V$ is not triangulable as a simplicial complex. However, by the work of Siebenmann and others, it is known that every orientable topological 5-dimensional closed manifold can be triangulated as a simplicial complex, see Theorem 21.5. This implies that $M_1= V \times S^1$ is triangulable, so $M_k=M_1 \times T^{k-1}$ is triangulable for all $k \geq 1$.

Yes, such examples do in fact exist.

  See this paper, Examples 21.4: There are topological manifolds that can be triangulated as simplicial complexes but do not admit any PL structure.

Yes, such examples do in fact exist. See Rudyak's paper Piecewise linear structures on topological manifolds, Examples 21.4:

There are topological manifolds that can be triangulated as simplicial complexes but do not admit any PL structure.

Such examples exist in fact in any dimension $n \geq 5$, and are of the form $$M_k=V \times T^k, \quad k \geq 1,$$ where $V$ is the famous $E_8$-manifold constructed by Friedman. See Theorem 7.2 and Corollary 7.4 in the quoted paper.

It is worth remarking that the $4$-manifold $V$ is not triangulable as a simplicial complex. However, by the work of Siebenmann and others, it is known that every orientable topological 5-dimensional closed manifold can be triangulated as a simplicial complex, see Theorem 21.5. This implies that $M_1= V \times S^1$ is triangulable, so $M_k=M_1 \times T^{k-1}$ is triangulable for all $k$.