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Expanding my comment: The natural map $H_*(X,A)\rightarrow H_*(X\cup_f Y, Y)$ is an isomorphism, this is the so-called "excision axiom". The map $\pi_i(X,A)\rightarrow \pi_i(X\cup_f Y, Y)$ is an iso …

The mapping torus $T$ of the complex-conjugation-map $\mathbb{C}P^2 \rightarrow \mathbb{C}P^2$ does the job. For example by running the Serre spectral sequence with local coefficients, you obtain th …

We have that $S^3 \simeq \Omega \mathbb{H}P^{\infty}$, so by adjointness we can as well consider the group of maps $[\Sigma S^3 \times S^3, \mathbb{H}P^{\infty}]$. It is well-known that $[X,\Omega Y] …

This is a modification of Anders's suggestion: Take $X=M(\mathbb{Z}[\frac{1}{2}],2)$ to be the Moore space with $H_2(X) = \mathbb{Z}[\frac{1}{2}]$. We can have $\mathbb{Z}/2$ act on $X$ this by a sig …

The question in the title differs from the question spelled out in the post: In the title, you ask for embedded manifolds, in the post you ask for just maps from manifolds. I think the version of the …

To add onto the very nice explicit answer of Tyrone, here's another perspective which you might find interesting: Both Stiefel-Whitney classes and Wu classes can be generalized to arbitrary $\mathbb{F …

In dimension 4, we have the following: Simply-connectedness implies that $H_1(M)=0$. The condition that $M$ be a rational homology sphere implies that $H_2(M), H_3(M)$ are finitely generated torsion g …

By replacing $f: X\to Y$ by a CW inclusion you can attack this with classical obstruction theory. I think the obstructions will lie in relative cohomology groups with local coefficients, $H^{n+1}(Y,X; …

To expand on my comment: As written (i.e. without requiring a map $f:X\to Y$), this is false in general. For an example, write $S^2$ as homogeneous space, $S^2=SU(2)/U(1)$. This exhibits $S^2$ as the …

The definition you've most likely encountered is the following: For maps $W\xrightarrow{f} X \xrightarrow{g} Y \xrightarrow{h} Z$, such that adjacent maps compose to $0$, $g$ extends to a map on th …

No. The reason this holds in the complex case (any degree $0$ map $S^1 \to S^1$ factors through $\exp: \mathbb{R}i\to S^1$) is that in that case the exponential map is a covering map, in particular a …

This is the classical $\mathrm{lim}^1$ phenomenon: While $\mathrm{Map}(X,Y) = \operatorname{lim}\mathrm{Map}(X_n,Y)$ (a homotopy limit), on homotopy classes it is not true that $[X,Y] \cong \operatorn …

Let me me write $ko$ for connective $KO$, since I'm more used to that. If $ko_*ko$ were a flat $ko_*$-module, then by basechange to $H\mathbb{Z}$, $H\mathbb{Z}_*ko$ would be a flat $\mathbb{Z}$-module …

The Lefschetz fixed point theorem implies that any $f: S^n \to S^n$ without fixed points has degree $(-1)^{n+1}$. But an even map $S^n \to S^n$ has even degree, since it factors as $$ S^n \xrightarrow …

Being a manifold or the dimension restriction $k\leq n$ doesn't matter, the following applies to finite simplicial complexes in general: As others have explained, if the fundamental group is not finit …