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One needs to distinguish between the direct sum and the direct product of a collection of groups. For a countably infinite collection of copies of $\mathbb Z$ the direct sum of these groups is a free …

Lebesgue numbers are certainly not needed to compute $\pi_1(S^1)$ using lifting properties of covering spaces. I checked eleven books that compute $\pi_1(S^1)$ using the covering space ${\mathbb R}\t …

If I understand the question correctly, you have a CW complex $Y'$ which is the union of two subcomplexes $Y$ and $X'$ whose intersection is the subcomplex $X$. We can first collapse $X$ to a point t …

These spaces $M_{p,q}=M_p\cup M_q$ are discussed in Examples 1.24 and 1.35 of my algebraic topology book. I don't know that they have a standard name, apart from $M_{2,2}$ which is the Klein bottle. …

Identifying opposite faces of a cube via a 90 degree twist gives a spherical manifold whose fundamental group is the quaternion group $\{\pm1,\pm i,\pm j,\pm k\}$. If one identifies opposite faces of …

$\newcommand{\Diff}{\mathrm{Diff}}$Probably the simplest such manifold is $S^1 \times S^2$, whose diffeomorphism group has the homotopy type of $O(2) \times O(3) \times \Omega SO(3)$. This has $\pi_2$ …

When thinking about chain homotopies in a setting involving simplices it can be helpful to consider the product $\Delta^p\times I$ where $\Delta^p$ is a $p$-simplex and $I=[0,1]$. The formula $h\sigm …

As suggested by Lennart Meier, the connected sum $M=P\#P$ of two copies of the Poincaré homology sphere gives an example. The homotopy groups $\pi_n(M)$ are nonzero for all $n>1$ because the universa …

The issue here is Dehn twists along curves parallel to the circles of $\partial S$. These usually generate infinite cyclic subgroups of ${\rm Mod}(S,Q)$, the only exceptions being when $S$ is a disk …

If ${\rm Diff}(M)$ is contractible then the question of course has a negative answer. Examples where this happens are known in dimension three but not in higher dimensions. For $M$ a closed hyperboli …

There exist closed orientable hyperbolic 3-manifolds that are surface bundles such that the fiber is the only incompressible surface in the manifold (up to isotopy). Such manifolds can be obtained by …

The authors of this book are attempting to use CW structures to justify certain cohomology isomorphisms, but this seems to be the wrong approach since some of their claims about CW structures are just …

In high dimensions ($\geq 5$) the most basic examples arise on manifolds with nonempty boundary, where one requires that diffeomorphisms restrict to the identity on the boundary. The simplest case is …

The quotient group $Diff_1(M)/Diff_0(M)$ is a discrete group since $ Diff_0(M)$ is a path component of $Diff(M)$, hence also a connected component since $Diff(M)$ is locally path-cconnected, and $Diff …

(This is a long comment rather than a complete answer.) As Igor Rivin points out, the mapping class group is not ${\mathbb Z}_2$. There is another ${\mathbb Z}_2$ direct summand coming from a homeomor …