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It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).

What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso on $\pi_1$, see e.g. this questionthis question.

(On the other hand, a map which is only a homology equivalence does not need to be an isomorphism on $\pi_1$ and hence not a weak equivalence.)

What is a concrete example of a map that is an isomorphism on homology with $\mathbb Z$ coefficients, and also an isomorphism on $\pi_1$, but not a weak equivalence?

It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).

What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso on $\pi_1$, see e.g. this question.

(On the other hand, a map which is only a homology equivalence does not need to be an isomorphism on $\pi_1$ and hence not a weak equivalence.)

What is a concrete example of a map that is an isomorphism on homology with $\mathbb Z$ coefficients, and also an isomorphism on $\pi_1$, but not a weak equivalence?

It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).

What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso on $\pi_1$, see e.g. this question.

(On the other hand, a map which is only a homology equivalence does not need to be an isomorphism on $\pi_1$ and hence not a weak equivalence.)

What is a concrete example of a map that is an isomorphism on homology with $\mathbb Z$ coefficients, and also an isomorphism on $\pi_1$, but not a weak equivalence?

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edit approved Feb 18, 2015 at 16:00
user62675
edit approved Feb 18, 2015 at 16:00
user62675

Homology equivalence and isoisomorphism on pi_1$\pi_1$ not enough for homotopy equivalence?

It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).

What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso on $\pi_1$, see e.g. Reference needed: Isomorphism on pi_1 and homology gives weak equivalencethis question.

(On the other hand, a map which is only a homology equivalence does not need to be an isoisomorphism on $\pi_1$ and hence not a weak equivalence.)

My question now is: What is a concrete example of a map that is an iso on homology with $\mathbb Z$ coefficients, and also an iso on $\pi_1$, but $not$ a weak equivalence?

What is a concrete example of a map that is an isomorphism on homology with $\mathbb Z$ coefficients, and also an isomorphism on $\pi_1$, but not a weak equivalence?

Homology equivalence and iso on pi_1 not enough for homotopy equivalence?

It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).

What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso on $\pi_1$, see e.g. Reference needed: Isomorphism on pi_1 and homology gives weak equivalence.

(On the other hand, a map which is only a homology equivalence does not need to be an iso on $\pi_1$ and hence not a weak equivalence.)

My question now is: What is a concrete example of a map that is an iso on homology with $\mathbb Z$ coefficients, and also an iso on $\pi_1$, but $not$ a weak equivalence?

Homology equivalence and isomorphism on $\pi_1$ not enough for homotopy equivalence?

It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).

What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso on $\pi_1$, see e.g. this question.

(On the other hand, a map which is only a homology equivalence does not need to be an isomorphism on $\pi_1$ and hence not a weak equivalence.)

What is a concrete example of a map that is an isomorphism on homology with $\mathbb Z$ coefficients, and also an isomorphism on $\pi_1$, but not a weak equivalence?

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Jens Reinhold
Jens Reinhold
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Homology equivalence and iso on pi_1 not enough for homotopy equivalence?

It is a standard consequence of Hurewicz's theorem that a homology eqivalence between simply connected spaces is a weak equivalence (and hence a homotopy equivalence, if the spaces are CW-complexes).

What is more, it is even enough to assume that the map is a homology equivalence with local coefficients and an iso on $\pi_1$, see e.g. Reference needed: Isomorphism on pi_1 and homology gives weak equivalence.

(On the other hand, a map which is only a homology equivalence does not need to be an iso on $\pi_1$ and hence not a weak equivalence.)

My question now is: What is a concrete example of a map that is an iso on homology with $\mathbb Z$ coefficients, and also an iso on $\pi_1$, but $not$ a weak equivalence?