# On $n$-connected simplicial complex and vanishing of reduced homology

Let $$X=|\Delta|$$ be the geometric realization of an abstract simplicial complex $$\Delta$$. Let $$k$$ be a field. Assume that $$X$$ is path connected .

Consider the following two conditions:

(1) $$\pi_i (X)=0, \forall 1\le i\le n$$

(2) $$\tilde H_i(X,k)=0, \forall 0\le i\le n$$

I know that (1) implies (2) by Hurewicz theorem and Universal coefficient theorem. My question is : Does (2) imply (1) ? If not, then would some other vanishing result on (reduced) Homology imply (1) ?

The Poincare Homology sphere has $$\tilde H_0$$, $$H_1$$ and $$H_2$$ trivial, yet has non-trivial $$\pi_1$$. But the Abelianisation of $$\pi_1$$ is trivial. I suspect that "(2) implies (1)" is far too optimistic, even if you replace $$\pi_1(X)$$ by its Abelianisation in (1).

• Thanks for the interesting example ... is it a simplicial complex ?
– user
Sep 12, 2019 at 22:47
• Every smooth manifold can be triangulated. In the wikipedia page there's a link to a particularly economical triangulation. Sep 13, 2019 at 3:01

If we use $$\mathbb Z$$-coefficients, then the Hurewicz theorem tells us that (2) implies (1) provided $$X$$ is simply connected. If it is not, then there are counterexamples (see Lord Shark the Unknown's answer).

If we use coefficients in a field $$k$$, then it is possible that (2) is satisfied, but $$\tilde{H}_i(X) \ne 0$$ for some $$i \le n$$. As an example take the space $$X$$ obtained by attaching to $$S^2$$ a cell $$D^3$$ via map $$f : S^2 \to S^2$$ of degree $$2$$. This space is simply connected and has $$\pi_2(X) = \mathbb Z_2$$. By the Hurewicz theorem $$H_2(X) = \mathbb Z_2$$ and by the universal coefficient theorem $$H_2(X,\mathbb Q) = 0$$. See An abelian group A is torsion iff A ⊗ Q = 0.

This example shows that (for $$k = \mathbb Q$$) even for simply connected $$X$$ (2) does not imply (1).

However, there exists a "rational Hurewicz theorem" which allows to get information about $$\pi_i(X) \otimes \mathbb Q$$. See https://en.wikipedia.org/wiki/Hurewicz_theorem.

• I thought Hurewicz theorem only says that vanishing of the homotopy groups imply vanishing of homology groups and NOT the other way around ...
– user
Sep 13, 2019 at 9:35
• The converse holds for simply connected spaces. See for example Theorem 7.5.5 in Spanier, Edwin H. Algebraic topology. Vol. 55. No. 1. Springer Science & Business Media, 1989. You can also derive it from the wikipedia-article. You can also use Hatcher Theorem 4.32 (prove the converse inductively). Sep 13, 2019 at 9:42