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To clarify one point in the previous answers of Jeff and Mark: There are two different definitions of "deformation retraction" that are often used. In the stronger notion the subspace has to be poin …

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 $ …

There are three possible definitions of an H-space, according to whether the "identity" element is a strict left and right identity, or only an identity up to basepoint-preserving homotopy, or just up …

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 …

Often the definition of a manifold includes a condition to exclude strange manifolds that are "very large" such as the Prüfer surface. One condition commonly assumed is that the topology on the manifo …

Here is an example of a $p$-sheeted covering space $X\to Y$ such that $H_1(X)$ has an element of order $p$ but $H_1(Y)$ is free. The space $X$ is the union of two pieces: a torus $A$ and surface $B$ o …

There are many $K(G,1)$ spaces with nicely computable homology groups. Besides the ones you listed, there are many compact 3-manifolds of this type. Also various configuration spaces such as the space …

This isn't really an answer but it's a bit long for a comment. Do you really mean cohomology rather than homology? The dual of a Postnikov tower is usually considered to be a homology decomposition, …

Let $X$ be the CW complex obtained from $S^1 \vee S^n$, $n>1$, by attaching an $(n+1)$-cell via a map $S^n\to S^1\vee S^n$ representing the element $2t-1$ in $\pi_n(S^1\vee S^n) \cong {\mathbb Z}[t,t^ …

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 …

In the other replies there has been some mention of alternative methods for subdivision besides barycentric subdivision, but these are rarely encountered in algebraic topology. What are some of these …

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 …

Here is a simple-minded example that is probably not the sort of thing you are looking for since it is not defined in terms of maps. For a topological space $X$ define its "local homology spectrum" to …

Some further comments on homology spheres: First, for any closed connected orientable $n$-manifold $M$ there is always a degree $1$ map $M\to S^n$ obtained by collapsing the complement of an open bal …

Here's an example that's a 2-dimensional CW complex. Start with a 0-cell, then attach three 1-cells labeled $a$, $b$, $c$ to get a wedge of three circles, then attach a 2-cell via the word $aba^{-1}b^ …