Are there compact manifolds homotopy equivalent to a wedge sum of compact manifolds?

Question

One example given by Hatcher as an application for the cohomology ring is to distinguish $\mathbb{CP}^2$ from $S^2 \vee S^4$ up to homotopy equivalence despite their cohomology groups being the same. But I would like to know if there are any examples, preferably low-dimensional, of compact manifolds that are homotopy equivalent to wedge sums of two or more compact manifolds.

Answer 1

A nontrivial wedge sum $M \vee N$ of closed manifolds fails to satisfy Poincaré duality, and so cannot be homotopy equivalent to another closed manifold $X$.

First, if $M$ and $N$ have the same dimension, then top (co)homology distinguishes $M \vee N$ from a closed manifold. So assume WLOG that $M$ has strictly smaller dimension than $N$. Recall that Poincaré duality over $\mathbb{F}_2$ implies that the cup product pairing

$$H^k(X, \mathbb{F}_2) \times H^{n-k}(X, \mathbb{F}_2) \to H^n(X, \mathbb{F}_2) \cong \mathbb{F}_2$$

is nondegenerate (here $n = \dim X = \dim N$). But taking $k = \dim M$, there is a class in $H^k(X, \mathbb{F}_2)$ coming from a generator of $H^k(M, \mathbb{F}_2)$ which does not cup nontrivially into any higher degree.