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Take any pseudoisotopy $\varphi\colon M\times I\rightarrow M\times I$ from the identity to a diffeomorphism $\phi$ that is not isotopic to the identity (as you mentioned, these exist). By obstruction …

Appendix B of Randal-Williams' "Cohomology of automorphism groups of free groups with twisted coefficients" gives a stable description of the graded $\mathbb{Q}[\Sigma_q]$-module $H^*(\Gamma_g;H^{\oti …

For many values of $n$, the answer to both questions is no. Since the fundamental groups of $BX$ and $B\mathrm{Diff}(S^n)$ are finite for $n\ge5$ (This uses that $\pi_0\mathrm{Diff}_\partial(D^n)$ is …

This is a follow-up to Dan Ramras' answer of this question. The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here). The weak Hausdorff rather t …

Let me assume that M is at least 5-dimensional. Sullivan's proof only uses surgery theory and properties of O(n) that also hold for Top(n), so the answer to your first question is yes. Regarding your …

For $n=1$, the answer to your question is negative, as explained by Gregory Arone in the comments. In the cases $n\neq 1,2,4$, there is the following easy argument: The long exact sequence of the fib …

The classification problem of smooth oriented closed $(n-1)$-connected $2n$-manifolds for $n\ge3$ splits into three parts. Classify smooth almost closed compact oriented $(n-1)$-connected $2n$-manifo …

Repeating the first part of Oscar's answer and elaborating on comments by Chris and Panagiotis, here is a down-to-earth argument in all cases: The cases $n=1,3,7$ are fine, since the stable stems are …

Formality of a space is meant in the sense of Sullivan, i.e. a space $X$ is called formal, if its commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is weak …

The mapping class group of a smooth manifold $M$ is the group of all its self diffeomorphisms up to isotopy, i.e. $\pi_0(\operatorname{Diff}(M))\cong \pi_1(B\operatorname{Diff}(M))$. A large portion …

Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces $$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction. Richard Pa …

Stiefel-Whitney numbers detect (unoriented) bordism classes and together with Pontryagin numbers, they determine oriented bordism classes. Ranicki's total surgery obstruction of a finite $n$-dimensio …

If everything takes place in the category of compactly generated spaces, it holds $$\pi_0(BC)=\pi_0(obC)/\tilde{},$$ where two path components in the object space get identified, if there are objects …

To avoid leaving this question open: Assuming we work in the category of compactly generated spaces, geometric realization commutes with pullbacks.(It's crucial that we use the compactly generated pr …

As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses. Let me present one case for which the question can be answered positive which is basical …