Manifolds with Euler characteristic equal to $\pm 1$

algebraic-topologycompact-manifoldshomology-cohomology

A compact connected smooth surface has Euler characteristic equal to $\pm 1$ if and only it is homeomorphic to the real projective plane or the connected sum of $3$ real projective planes. What are other examples of connected manifolds (of even dimension) that have Euler characteristic equal to $\pm 1$? For instance, one could take products of the above surfaces.

Best Answer

The Euler characteristic of a closed orientable manifold of dimension $4k + 2$ is even, so to find orientable examples, you need to look in dimensions which are a multiple of four.

One infinite collection of examples can be found amongst manifolds of the form $M_{a,b} := a\mathbb{CP}^2\#b(S^1\times S^3)$, i.e. the connected sum of $a$ copies of $\mathbb{CP}^2$ and $b$ copies of $S^1\times S^3$. Such a manifold has Euler characteristic $2 + a - 2b$, so $\chi(M_{2b-1,b}) = 1$ for any $b \geq 1$, and $\chi(M_{2b-3, b}) = -1$ for any $b \geq 2$. You can vary this construction in many ways:

  • replace some of the $\mathbb{CP}^2$'s with $\overline{\mathbb{CP}^2}$'s,
  • replace $S^1\times S^3$ with any manifold with Euler characteristic zero, e.g. $T^4$,
  • go to dimension $n = 8$ or $16$ where you can replace $\mathbb{CP}^2$ with $\mathbb{HP}^2$ or $\mathbb{OP}^2$ respectively, and replace $S^1\times S^3$ with $S^1\times S^{n-1}$.

Because the Euler characteristic is multiplicative, given any two manifolds with Euler characteristic $\pm 1$, their product also has Euler characteristic $\pm 1$. In particular, $M_{1,1}^k = (\mathbb{CP}^2\#(S^1\times S^3))^k$ gives an example of a closed orientable $4k$-manifold with Euler characteric $1$. More generally, if $F \to E \to B$ is a fiber bundle and $\chi(F), \chi(B) \in \{-1, 1\}$, then $\chi(E) = \chi(B)\chi(F) \in \{-1,1\}$.

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