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(This answer has been edited to give more details.) Finitely generated homotopy groups do not imply finitely generated homology groups. Stallings gave an example of a finitely presented group $G$ suc …

Take the composition of a degree one map $f:T^3\to S^3$ with the Hopf map $g:S^3\to S^2$, where $T^3$ is the 3-torus. This composition is trivial on homotopy groups since $T^3$ is aspherical and $\pi_ …

Simplicial sets and simplicial complexes lie at two ends of a spectrum, with Delta complexes, which were invented by Eilenberg and Zilber under the name "semi-simplicial complexes", lying somewhere in …

The elementary proof that $\pi_1$ is abelian applies more generally to H-spaces (spaces $X$ with a continuous multiplication map $X \times X \to X$ having a 2-sided identity element) without any assum …

It is easy to understand the existence of a Thom class by considering cellular cohomology. Let the given vector bundle be $E\to B$ with fibers of dimension $n$. One can assume without significant lo …

Two simple examples with lots of interesting differentials are given by the Serre spectral sequences for integer homology (rather than cohomology) for the fibrations $$K({\mathbb Z}_2,1) \to K({\mathb …

$\newcommand{\Diff}{\mathrm{Diff}}$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 $\pi_2$ …

Homology also has complicated and unintuitive aspects if one goes beyond nice spaces like CW complexes. A surprising example of this is the subspace of Euclidean 3-space consisting of the union of a c …

This is indeed a mystery. I presume the question refers to wedges of spheres of the same dimension, where there's a simple criterion (n-dimensional and (n-1)-connected, for some n). For wedges of sph …

Here is a method for constructing examples. If a fiber bundle $F \to E \to B$ has a section, the associated long exact sequence of homotopy groups splits, so the homotopy groups of $E$ are the same a …

While my Algebraic Topology book and my unfinished book on spectral sequences (referred to in other answers to this question) contain some information about homotopy groups of spheres, they don't real …

The basic fact is that the 2-torsion all has order exactly 2, so it injects into the mod 2 cohomology, forming a subalgebra of the polynomial algebra on the Stiefel-Whitney classes. This subalgebra c …

The earlier answers showing that the fundamental group of this space is infinite cyclic by determining its universal cover or by constructing a fiber bundle over it with contractible fibers are very n …

One can compute $\pi_3$ completely by using the fact that it is isomorphic to $\pi_3$ of the universal cover. For a connected sum $M=P\# Q$ of lens spaces $P$ and $Q$ one obtains the universal cover $ …

Here is an example where ${\rm Diff}(M)$ with the compact-open topology is not homotopy equivalent to a CW complex. Take $M$ to be a surface of infinite genus, say the simplest one with just one nonco …