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I know that the converse of this statement is true but I am not sure how to go about finding out the answer to this question.

Asvin
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    Have you thought through the case of trivial fundamental groups? Does the conclusion hold then? – hardmath Dec 10 '13 at 04:26
  • Related: http://math.stackexchange.com/q/1901/ – Grigory M Dec 10 '13 at 15:50
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    I'm not sure why this question is receiving down votes. It sounds to me like the OP has just been introduced to the fundamental group and is asking a very natural question which might only seem obvious to those with more experience. – Dan Rust Dec 10 '13 at 16:10
  • @DanielRust I concur. This is a perfectly valid question. In particular, it can serve as a motivation for homology. – Ayman Hourieh Dec 10 '13 at 17:22

2 Answers2

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Assuming you're taking a course at the moment in algebraic topology, you will soon be (or have already been) introduced to the homology of a space which is another (family of) homotopy invariant of topological spaces, in the same spirit of the fundamental groups. With this invariant, you will find that the first homology groups of the one-point space $\{0\}$ and the $2$-sphere $S^2$ give $H_2(\{0\})=0$ and $H_2(S^2)\cong \mathbb{Z}$, which implies that $S^2$ and the single point $\{0\}$ are not homotopy equivalent, even though they have isomorphic (both trivial) fundamental groups.

This is a recurring theme. Although the fundamental group can distinguish many spaces up to homotopy, it is not a total invariant. It's just one of many that have been defined and studied by those working in algebraic topology and abstract homotopy theory. You can think of the fundamental group as a good first step for trying to distinguish two spaces, but should be used in conjunction with the many other, more powerful invariants of the subject. The real power of the fundamental group, in many cases, is that it's readily computable and often easy to visualise.

Dan Rust
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  • In that case, is there some sort of general converse? Maybe all the homology groups together uniquely determine the space upto homotopy? – Asvin Dec 16 '13 at 13:22
  • Fortunately not (otherwise many homotopy theorists would be out of a job). It's a little more difficult to cook up an example of two spaces with isomorphic fundamental groups and all (co)homology groups, but they do exist. In general, we need a whole arsenal of invariants to fire at two arbitrary topological spaces in order to distinguish them up to homotopy and there have been hundreds of thousands (millions?) of pages written on trying to increase this arsenal. – Dan Rust Dec 16 '13 at 22:42
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Compare $\mathbb{S}^2$ and $\mathbb{R}^2$.

Neal
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