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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 …

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 …

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] …

No and no. For an explicit counterexample to 1. (which is also a counterexample to 2.) take the map $\mathbb{R}P^2\to \mathbb{R}P^{\infty}$.

This holds only for compact objects (i.e. finite CW spectra), since it is easy to see that additivity fails otherwise (the other axioms of homology theories are satisfied). The usual way to obtain a h …

Both the Atiyah-Hirzebruch and the Serre spectral sequence can be constructed from a skeletal filtration of a CW-complex: We can construct the Atiyah-Hirzebruch spectral sequence by filtering $X$ by …

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 …

I know this post is quite old, but in case you are still interested, or anyone else is, I thought about sharing my recent thoughts about the topic. After all, this is the second result on "matric toda …

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 …

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 …

Here's a direct way to relate the two: One more structure is additivity of orientations. For $V, W$ of dimensions $n,m$, we have a canonical pairing $$ \Lambda^n V \otimes \Lambda^m W \cong \Lambda^{n …

This is true for finite $\pi_1$ and false for infinite $\pi_1$: Let $\widetilde{X}$ denote the universal cover of $X$, then $\Omega\widetilde{X}$ is the unit connected component of $\Omega X$, and $\O …

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 …

An $E_{\infty}$ structure extending the $E_1$ structure on $R(n)$ in particular yields maps $(\mathbb{S}^{2n})^{\otimes p}_{hC_p} \to \mathbb{S}^{2pn}$ splitting the inclusion of the bottom cell. The …

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 …