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I know of several different arguments. You can decide which one you think is most elegant... Rohlin's argument, which is actually quite geometric. You start with an immersion of the 3-manifold in $ …

If $S$ is a closed hyperbolic surface, is there an easy proof of the injectivity of the Dehn-Nielsen-Baer map from $\mathrm{Mod}(S)$ to $\mathrm{Out}(\pi_1(S))$, taking an element of the mapping class …

If $\Sigma_1$ and $\Sigma_2$ are two compact topological surfaces with boundary and $\phi, \psi : \Sigma_1 \hookrightarrow \Sigma_2$ are two orientation-preserving embeddings that are homotopic, then …

Does anyone have good examples of a space $X$ and a map $f: X \to X$ so that $f_*: H_*(X) \to H_*(X)$ is the identity but (e.g.) $f_*: H_*(X; \mathbb{F}_2) \to H_*(X; \mathbb{F}_2)$ is not the identit …

Under what circumstances is a quasi-isomorphism between two complexes necessarily a homotopy equivalence? For instance, this is true for chain complexes over a field (which are all homotopy equivalen …

With your definitions, assuming you mean the configuration space of distinct points, and that the inclusions need to be compatible with the actual locations of the points in some way, there is not suc …

For a more explicit example than Chris's, consider the map from the (2-dimensional) torus to a sphere that collapses the 1-skeleton of the usual CW complex and takes the 2-cell to the 2-cell of the sp …

(1) and (2) are perfectly compatible, to the extent which (1) makes sense. $\mathbb{R}P^k$ is naturally a CW complex with one cell in each dimension, and $\mathbb{R}P^{k-2}$ is a subcomplex. Assuming …

$\widetilde{\textit{SL}_2(\mathbb{R})}$ is not so complicated, but one of the best descriptions is just that. It has no faithful finite-dimensional representations, which makes things a little tricky …

Take a contraction of $x$ in $A$. That gives an $n$-chain $y_A$ living in $A$, whose boundary is $x$. Similarly, a contraction of $x$ in $B$ gives an $n$-chain $y_B$, again with boundary equal to $x …

This is an interesting line of questions, but I think it doesn't quite work as stated. First off, your notion of "homotopy" is not an equivalence relation (as far as I understand it), so it won't agr …

A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are …

Any continuous map from M to N is homotopic to a smooth map, and if two smooth maps are homotopic, then they are also smoothly homotopic. (More generally, two homotopic functions are homotopic throug …

Torsten's answer was good, but there are also more elementary answers. Here's one, which is essentially a big transversality argument, followed by a mild de-singularization. Let me consider $M$ to b …