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name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #8 to #9: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='homological_algebra'>Homological algebra</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/homological+algebra'>homological algebra</a></strong></p> <p>(also <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+homological+algebra'>nonabelian homological algebra</a>)</p> <p><em><a class='existingWikiWord' href='/schreiber/show/diff/Introduction+to+Homological+Algebra' title='schreiber'>Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/additive+and+abelian+categories'>additive and abelian categories</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Ab-enriched+category'>Ab-enriched category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pre-additive+category'>pre-additive category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/additive+category'>additive category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pre-abelian+category'>pre-abelian category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+category'>abelian category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+category'>Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf'>abelian sheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/semi-abelian+category'>semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/kernel'>kernel</a>, <a class='existingWikiWord' href='/nlab/show/diff/cokernel'>cokernel</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/complex'>complex</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential'>differential</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+of+chain+complexes'>category of chain complexes</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/chain+map'>chain map</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/chain+homotopy'>chain homotopy</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>chain homology and cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homological+resolution'>homological resolution</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cochain+on+a+simplicial+set'>simplicial homology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+homology'>generalized homology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>exact sequence</a>,</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>short exact sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>long exact sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/split+exact+sequence'>split exact sequence</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/injective+object'>injective object</a>, <a class='existingWikiWord' href='/nlab/show/diff/projective+object'>projective object</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>injective resolution</a>, <a class='existingWikiWord' href='/nlab/show/diff/projective+resolution'>projective resolution</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/flat+resolution'>flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+category'>derived category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/triangulated+category'>triangulated category</a>, <a class='existingWikiWord' href='/nlab/show/diff/enhanced+triangulated+category'>enhanced triangulated category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+model+category'>stable model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pretriangulated+dg-category'>pretriangulated dg-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-category'>A-∞-category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+chain+complexes'>(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functor</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+functor+in+homological+algebra'>derived functor in homological algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tor'>Tor</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lim%5E1+and+Milnor+sequences'>lim^1 and Milnor sequences</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fr-code'>fr-code</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/double+complex'>double complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Koszul-Tate+resolution'>Koszul-Tate resolution</a>, <a class='existingWikiWord' href='/nlab/show/diff/BV-BRST+formalism'>BRST-BV complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence'>spectral sequence</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence+of+a+filtered+complex'>spectral sequence of a filtered complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence+of+a+double+complex'>spectral sequence of a double complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+spectral+sequence'>Grothendieck spectral sequence</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Leray+spectral+sequence'>Leray spectral sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Serre+spectral+sequence'>Serre spectral sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hochschild-Serre+spectral+sequence'>Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/diagram+chasing'>diagram chasing</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/3x3+lemma'>3x3 lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/four+lemma'>four lemma</a>, <a class='existingWikiWord' href='/nlab/show/diff/five+lemma'>five lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/snake+lemma'>snake lemma</a>, <a class='existingWikiWord' href='/nlab/show/diff/connecting+homomorphism'>connecting homomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/horseshoe+lemma'>horseshoe lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Baer%27s+criterion'>Baer's criterion</a></p> </li> </ul> <p><a class='existingWikiWord' href='/nlab/show/diff/Schanuel%27s+lemma'>Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/singular+homology'>singular homology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cyclic+homology'>cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a> / <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal</a>, <a class='existingWikiWord' href='/nlab/show/diff/operadic+Dold-Kan+correspondence'>operadic</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Moore+complex'>Moore complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Alexander-Whitney+map'>Alexander-Whitney map</a>, <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Zilber+map'>Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Zilber+theorem'>Eilenberg-Zilber theorem</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cochain+on+a+simplicial+set'>cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+coefficient+theorem'>universal coefficient theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K%C3%BCnneth+theorem'>Künneth theorem</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#for_long_cohomology_exact_sequences'>For long (co)homology exact sequences</a><ul><li><a href='#OnHomologyInTermsOfElements'>In terms of elements</a></li><li><a href='#OnHomologyGeneralAbstract'>General abstract</a></li></ul></li><li><a href='#examples'>Examples</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#relation_to_homotopy_fiber_sequences'>Relation to homotopy fiber sequences</a></li></ul></li><ins class='diffins'><li><a href='#related_concepts'>Related concepts</a></li></ins><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>Generally, a <em>connecting homomorphism</em> is a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> of the kind produced by the <a class='existingWikiWord' href='/nlab/show/diff/snake+lemma'>snake lemma</a>.</p> <p>Specifically, when the <a class='existingWikiWord' href='/nlab/show/diff/double+complex'>double complex</a> that goes into the snake lemma is regarded as part of a <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>short exact sequence</a> <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>B</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>A_\bullet \to B_\bullet \to C_\bullet</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complexes</a>, then the connecting homomorphisms induce morphisms <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>δ</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\delta_n : H_n(C) \to H_{n-1}(A)</annotation></semantics></math> on the <a class='existingWikiWord' href='/nlab/show/diff/homology'>homology groups</a> of these chain complexes which exhibit the corresponding <a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+homology'>long exact sequence in homology</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><msub><mi>δ</mi> <mi>n</mi></msub></mrow></mover><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>⋯</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \cdots \to H_n(A) \to H_n(B) \to H_n(C) \stackrel{\delta_n}{\to} H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C) \to \cdots \,. </annotation></semantics></math></div> <p>This long exact sequence is the image under <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>chain homology</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'> H_0(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A} </annotation></semantics></math></div> <p>of the long <a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>homotopy fiber sequence</a> of chain complexes induced by the short exact sequence. Hence the connecting homomorphism is the image under <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_\bullet(-)</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a> inclusion on chain complexes.</p> <h2 id='for_long_cohomology_exact_sequences'>For long (co)homology exact sequences</h2> <p>In the case that <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi><mo>≃</mo><mi>R</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A} \simeq R</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/Mod'>Mod</a> for some <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a> <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>, the construction of the connecting homomorphism for <a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+homology'>homology long exact sequences</a> is easily described in terms of elements and checking its properties is elementary, see <em><a href='#OnHomologyInTermsOfElements'>In terms of elements</a></em> below. By the <a href='abelian%20category#EmbeddingTheorems'>embedding theorems</a> the general case can be reduced to this case. But there is also a general abstract description without recourse to elements, which we discuss further below in <em><a href='#OnHomologyGeneralAbstract'>General abstract construction</a></em> .</p> <h3 id='OnHomologyInTermsOfElements'>In terms of elements</h3> <p>Let <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/ring'>commutative ring</a> and let <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi><mo>=</mo><mi>R</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A} = R</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/Mod'>Mod</a>. Write <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/category+of+chain+complexes'>category of chain complexes</a> in <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>.</p> <p>Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>A</mi> <mo>•</mo></msub><mover><mo>→</mo><mi>i</mi></mover><msub><mi>B</mi> <mo>•</mo></msub><mover><mo>→</mo><mi>p</mi></mover><msub><mi>C</mi> <mo>•</mo></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'> 0 \to A_\bullet \stackrel{i}{\to} B_\bullet \stackrel{p}{\to} C_\bullet \to 0 </annotation></semantics></math></div> <p>be a <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>short exact sequence</a> in <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>𝒜</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch_\bullet(\mathcal{A})</annotation></semantics></math>.</p> <div class='num_defn' id='ConnectingForHomologyInComponents'> <h6 id='definition'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{Z}</annotation></semantics></math>, define a <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>group homomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>δ</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \delta_n : H_n(C) \to H_{n-1}(A) \,, </annotation></semantics></math></div> <p>called the <strong><math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th connecting homomorphism</strong> of the short exact sequence, by sending</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>δ</mi> <mi>n</mi></msub><mo>:</mo><mo stretchy='false'>[</mo><mi>c</mi><mo stretchy='false'>]</mo><mo>↦</mo><mo stretchy='false'>[</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \delta_n : [c] \mapsto [\partial^B \hat c]_A \,, </annotation></semantics></math></div> <p>where</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c \in Z_n(C)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/cycle'>cycle</a> representing a given <a class='existingWikiWord' href='/nlab/show/diff/homology'>homology group</a>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>∈</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\hat c \in C_n(B)</annotation></semantics></math> is any lift of that cycle to an element in <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>B_n</annotation></semantics></math>, which exists because <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/surjection'>surjection</a> (but which no longer needs to be a cycle itself);</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>[\partial^B \hat c]_A</annotation></semantics></math> is the <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>-homology class of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\partial^B \hat c</annotation></semantics></math> which is indeed in <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>↪</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>A_{n-1} \hookrightarrow B_{n-1}</annotation></semantics></math> by exactness (since <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo stretchy='false'>)</mo><mo>=</mo><msup><mo>∂</mo> <mi>C</mi></msup><mi>p</mi><mo stretchy='false'>(</mo><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo stretchy='false'>)</mo><mo>=</mo><msup><mo>∂</mo> <mi>C</mi></msup><mi>c</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>p(\partial^B \hat c) = \partial^C p(\hat c) = \partial^C c = 0</annotation></semantics></math>) and indeed in <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>↪</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>Z_{n-1}(A) \hookrightarrow A_{n-1}</annotation></semantics></math> since <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∂</mo> <mi>A</mi></msup><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>=</mo><msup><mo>∂</mo> <mi>B</mi></msup><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\partial^A \partial^B \hat c = \partial^B \partial^B \hat c = 0</annotation></semantics></math>.</p> </li> </ol> </div> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>Def. <a class='maruku-ref' href='#ConnectingForHomologyInComponents'>1</a> is indeed well defined in that the given map is independent of the choice of lift <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\hat c</annotation></semantics></math> involved and in that the group structure is respected.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>To see that the constructon is well-defined, let <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>˜</mo></mover><mo>∈</mo><msub><mi>B</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\tilde c \in B_{n}</annotation></semantics></math> be another lift. Then <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>−</mo><mover><mi>c</mi><mo stretchy='false'>˜</mo></mover><mo stretchy='false'>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>p(\hat c - \tilde c) = 0</annotation></semantics></math> and hence <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>−</mo><mover><mi>c</mi><mo stretchy='false'>˜</mo></mover><mo>∈</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>B</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\hat c - \tilde c \in A_n \hookrightarrow B_n</annotation></semantics></math>. This exhibits a homology-equivalence <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub><mo>≃</mo><mo stretchy='false'>[</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>˜</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>[\partial^B\hat c]_A \simeq [\partial^B \tilde c]_A</annotation></semantics></math> since <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∂</mo> <mi>A</mi></msup><mo stretchy='false'>(</mo><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>−</mo><mover><mi>c</mi><mo stretchy='false'>˜</mo></mover><mo stretchy='false'>)</mo><mo>=</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>−</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>˜</mo></mover></mrow><annotation encoding='application/x-tex'> \partial^A(\hat c - \tilde c) = \partial^B \hat c - \partial^B \tilde c</annotation></semantics></math>.</p> <p>To see that <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>δ</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\delta_n</annotation></semantics></math> is a group homomorphism, let <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>c</mi><mo stretchy='false'>]</mo><mo>=</mo><mo stretchy='false'>[</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>]</mo><mo>+</mo><mo stretchy='false'>[</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[c] = [c_1] + [c_2]</annotation></semantics></math> be a sum. Then <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>≔</mo><msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover> <mn>1</mn></msub><mo>+</mo><msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\hat c \coloneqq \hat c_1 + \hat c_2</annotation></semantics></math> is a lift and by linearity of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∂</mo></mrow><annotation encoding='application/x-tex'>\partial</annotation></semantics></math> we have <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub><mo>=</mo><mo stretchy='false'>[</mo><msup><mo>∂</mo> <mi>B</mi></msup><msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover> <mn>1</mn></msub><mo stretchy='false'>]</mo><mo>+</mo><mo stretchy='false'>[</mo><msup><mo>∂</mo> <mi>B</mi></msup><msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover> <mn>2</mn></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\partial^B \hat c]_A = [\partial^B \hat c_1] + [\partial^B \hat c_2]</annotation></semantics></math>.</p> </div> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>Under <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>chain homology</a> <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_\bullet(-)</annotation></semantics></math> the morphisms in the short exact sequence together with the connecting homomorphisms yield the <a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+homology'>homology long exact sequence</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><msub><mi>δ</mi> <mi>n</mi></msub></mrow></mover><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>⋯</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \cdots \to H_n(A) \to H_n(B) \to H_n(C) \stackrel{\delta_n}{\to} H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C) \to \cdots \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>Consider first the exactness of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo></mrow></mover><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>p</mi><mo stretchy='false'>)</mo></mrow></mover><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_n(A) \stackrel{H_n(i)}{\to} H_n(B) \stackrel{H_n(p)}{\to} H_n(C)</annotation></semantics></math>.</p> <p>It is clear that if <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>∈</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>↪</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>a \in Z_n(A) \hookrightarrow Z_n(B)</annotation></semantics></math> then the image of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>a</mi><mo stretchy='false'>]</mo><mo>∈</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[a] \in H_n(B)</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo><mo>=</mo><mn>0</mn><mo>∈</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[p(a)] = 0 \in H_n(C)</annotation></semantics></math>. Conversely, an element <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>b</mi><mo stretchy='false'>]</mo><mo>∈</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[b] \in H_n(B)</annotation></semantics></math> is in the kernel of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>p</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_n(p)</annotation></semantics></math> if there is <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>c \in C_{n+1}</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∂</mo> <mi>C</mi></msup><mi>c</mi><mo>=</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\partial^C c = p(b)</annotation></semantics></math>. Since <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is surjective let <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>∈</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>\hat c \in B_{n+1}</annotation></semantics></math> be any lift, then <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>b</mi><mo stretchy='false'>]</mo><mo>=</mo><mo stretchy='false'>[</mo><mi>b</mi><mo>−</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[b] = [b - \partial^B \hat c]</annotation></semantics></math> but <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mi>b</mi><mo>−</mo><msup><mo>∂</mo> <mi>B</mi></msup><mi>c</mi><mo stretchy='false'>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>p(b - \partial^B c) = 0</annotation></semantics></math> hence by exactness <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>−</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>∈</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>↪</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>b - \partial^B \hat c \in Z_n(A) \hookrightarrow Z_n(B)</annotation></semantics></math> and so <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>b</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[b]</annotation></semantics></math> is in the image of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_n(A) \to H_n(B)</annotation></semantics></math>.</p> <p>It remains to see that</p> <ol> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/image'>image</a> of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_n(B) \to H_n(C)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/kernel'>kernel</a> of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>δ</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\delta_n</annotation></semantics></math>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/kernel'>kernel</a> of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_{n-1}(A) \to H_{n-1}(B)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/image'>image</a> of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>δ</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\delta_n</annotation></semantics></math>.</p> </li> </ol> <p>This follows by inspection of the formula in def. <a class='maruku-ref' href='#ConnectingForHomologyInComponents'>1</a>. We spell out the first one:</p> <p>If <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>c</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[c]</annotation></semantics></math> is in the image of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_n(B) \to H_n(C)</annotation></semantics></math> we have a lift <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\hat c</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\partial^B \hat c = 0</annotation></semantics></math> and so <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>δ</mi> <mi>n</mi></msub><mo stretchy='false'>[</mo><mi>c</mi><mo stretchy='false'>]</mo><mo>=</mo><mo stretchy='false'>[</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\delta_n[c] = [\partial^B \hat c]_A = 0</annotation></semantics></math>. Conversely, if for a given lift <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\hat c</annotation></semantics></math> we have that <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>[\partial^B \hat c]_A = 0</annotation></semantics></math> this means there is <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>∈</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>a \in A_n</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∂</mo> <mi>A</mi></msup><mi>a</mi><mo>≔</mo><msup><mo>∂</mo> <mi>B</mi></msup><mi>a</mi><mo>=</mo><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\partial^A a \coloneqq \partial^B a = \partial^B \hat c</annotation></semantics></math>. But then <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>˜</mo></mover><mo>≔</mo><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>−</mo><mi>a</mi></mrow><annotation encoding='application/x-tex'>\tilde c \coloneqq \hat c - a</annotation></semantics></math> is another possible lift of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> for which <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∂</mo> <mi>B</mi></msup><mover><mi>c</mi><mo stretchy='false'>˜</mo></mover><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\partial^B \tilde c = 0</annotation></semantics></math> and so <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>c</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[c]</annotation></semantics></math> is in the image of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H_n(B) \to H_n(C)</annotation></semantics></math>.</p> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>Of course the situation for <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>cochain cohomology</a> is formally dual to this situation. For convenience we repeat the statement for dual chains:</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mo>•</mo></msup><mo>→</mo><msup><mi>B</mi> <mo>•</mo></msup><mo>→</mo><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding='application/x-tex'>A^\bullet \to B^\bullet \to C^\bullet</annotation></semantics></math> be a short exact sequence of cochain complexes.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>c</mi><msub><mo stretchy='false'>]</mo> <mi>C</mi></msub><mo>∈</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[c]_C \in H^n(C)</annotation></semantics></math> the class of a closed element <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math>, by surjectivity of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>B \to C</annotation></semantics></math> there is an element <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>∈</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>\hat c \in B</annotation></semantics></math> mapping to it. This need not be closed anymore, but of course <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>d</mi> <mi>B</mi></msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>d_B \hat c</annotation></semantics></math> is. By the fact that <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>B \to C</annotation></semantics></math> is a chain map we have that the image of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>d</mi> <mi>B</mi></msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>d_B \hat c</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> vanishes. Therefore by the exactness of the sequence the element <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>d</mi> <mi>B</mi></msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>d_B \hat c</annotation></semantics></math> may be regarded as a closed element of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. The cohomology class <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>d</mi> <mi>B</mi></msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>[d_B \hat c]_A</annotation></semantics></math> of this is what the connecting homomorphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>δ</mi> <mi>n</mi></msup><mo>:</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \delta^n : H^n(C) \to H^{n+1}(A) </annotation></semantics></math></div> <p>assigns to <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>c</mi><msub><mo stretchy='false'>]</mo> <mi>C</mi></msub></mrow><annotation encoding='application/x-tex'>[c]_C</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>δ</mi><mo>:</mo><mo stretchy='false'>[</mo><mi>c</mi><msub><mo stretchy='false'>]</mo> <mi>C</mi></msub><mo>↦</mo><mo stretchy='false'>[</mo><msub><mi>d</mi> <mi>B</mi></msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \delta : [c]_C \mapsto [d_B\hat c]_A \,. </annotation></semantics></math></div> <p>This is indeed well defined, in that it is independent of the choice of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\hat c</annotation></semantics></math>: for <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>′</mo></mrow><annotation encoding='application/x-tex'>\hat c'</annotation></semantics></math> another choice, we have that the difference <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>−</mo><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>′</mo></mrow><annotation encoding='application/x-tex'>\hat c - \hat c'</annotation></semantics></math> is in the kernel of <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>B \to C</annotation></semantics></math> hence is in <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. Then <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>d</mi> <mi>B</mi></msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>′</mo><mo>=</mo><msub><mi>d</mi> <mi>B</mi></msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>+</mo><msub><mi>d</mi> <mi>A</mi></msub><mo stretchy='false'>(</mo><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>−</mo><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d_B \hat c' = d_B \hat c + d_A(\hat c - \hat c')</annotation></semantics></math>. Hence <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>d</mi> <mi>B</mi></msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub><mo>=</mo><mo stretchy='false'>[</mo><msub><mi>d</mi> <mi>B</mi></msub><mover><mi>c</mi><mo stretchy='false'>^</mo></mover><mo>′</mo><msub><mo stretchy='false'>]</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>[d_B \hat c]_A = [d_B \hat c']_A</annotation></semantics></math>.</p> </div> <h3 id='OnHomologyGeneralAbstract'>General abstract</h3> <div class='num_theorem'> <h6 id='theorem'>Theorem</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>A</mi> <mo>•</mo></msub><mover><mo>→</mo><mi>f</mi></mover><msub><mi>B</mi> <mo>•</mo></msub><mover><mo>→</mo><mi>g</mi></mover><msub><mi>C</mi> <mo>•</mo></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>0 \to A_\bullet \stackrel{f}{\to} B_\bullet \stackrel{g}{\to} C_\bullet \to 0</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>short exact sequence</a> of <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complexes</a> in some <a class='existingWikiWord' href='/nlab/show/diff/abelian+category'>abelian category</a> <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding='application/x-tex'>\mathcal{A}</annotation></semantics></math>. Then for all <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{Z}</annotation></semantics></math> there are natural <em>connecting homomorphisms</em> <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∂</mo><mo>:</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\partial : H_n(C) \to H_{n-1}(A)</annotation></semantics></math> such that we have a <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>long exact sequence</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mover><mo>→</mo><mi>g</mi></mover><msub><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mi>f</mi></mover><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mi>g</mi></mover><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mi>f</mi></mover><mi>⋯</mi></mrow><annotation encoding='application/x-tex'> \cdots \stackrel{g}{\to} H_{n+1}(C) \stackrel{\partial}{\to} H_n(A) \stackrel{f}{\to} H_n(B) \stackrel{g}{\to} H_n(C) \stackrel{\partial}{\to} H_{n-1}(A) \stackrel{f}{\to} \cdots </annotation></semantics></math></div> <p>in <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>chain homology</a>.</p> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>Applying the <a class='existingWikiWord' href='/nlab/show/diff/snake+lemma'>snake lemma</a> to the <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>commuting diagram</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mn>0</mn></mtd> <mtd /> <mtd><mn>0</mn></mtd> <mtd /> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Z</mi> <mi>n</mi></msub><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Z</mi> <mi>n</mi></msub><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Z</mi> <mi>n</mi></msub><mi>C</mi></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>A</mi> <mi>n</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>B</mi> <mi>n</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>C</mi> <mi>n</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>d</mi></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>d</mi></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>d</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mfrac><mrow><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mrow><mi>im</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>A</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow></mfrac></mtd> <mtd><mo>→</mo></mtd> <mtd><mfrac><mrow><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mrow><mi>im</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>B</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow></mfrac></mtd> <mtd><mo>→</mo></mtd> <mtd><mfrac><mrow><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><mrow><mi>im</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow></mfrac></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mn>0</mn></mtd> <mtd /> <mtd><mn>0</mn></mtd> <mtd /> <mtd><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && 0 && 0 && 0 \\ && \downarrow && \downarrow && \downarrow \\ 0 &\to& Z_n A &\to& Z_n B &\to & Z_n C \\ && \downarrow && \downarrow && \downarrow \\ 0 &\to& A_n &\to& B_n &\to & C_n &\to & 0 \\ && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} \\ 0 &\to& A_{n-1} &\to& B_{n-1} &\to & C_{n-1} &\to& 0 \\ && \downarrow && \downarrow && \downarrow \\ && \frac{A_{n-1}}{im(d)(A_n)} &\to& \frac{B_{n-1}}{im(d)(B_n)} &\to & \frac{C_{n-1}}{im(d)(C_n)} &\to & 0 \\ && \downarrow && \downarrow && \downarrow \\ && 0 && 0 && 0 } </annotation></semantics></math></div> <p>shows that the rows in the commuting diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mfrac><mrow><msub><mi>A</mi> <mi>n</mi></msub></mrow><mrow><mi>im</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo></mrow></mfrac></mtd> <mtd><mo>→</mo></mtd> <mtd><mfrac><mrow><msub><mi>B</mi> <mi>n</mi></msub></mrow><mrow><mi>im</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo></mrow></mfrac></mtd> <mtd><mo>→</mo></mtd> <mtd><mfrac><mrow><msub><mi>C</mi> <mi>n</mi></msub></mrow><mrow><mi>im</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo></mrow></mfrac></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>d</mi></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>d</mi></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>d</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><msub><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><msub><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && \frac{A_{n}}{im(d)(A_{n+1})} &\to& \frac{B_{n}}{im(d)(B_{n+1})} &\to & \frac{C_{n}}{im(d)(C_{n+1})} &\to & 0 \\ && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{d}} \\ 0 &\to& Z_{n-1} A &\stackrel{f}{\to}& Z_{n-1} B &\stackrel{g}{\to}& Z_{n-1} C } </annotation></semantics></math></div> <p>are <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>exact sequences</a>. Therefore applying the <a class='existingWikiWord' href='/nlab/show/diff/snake+lemma'>snake lemma</a> to this, once more, yields the desired long exact sequence.</p> </div> <h2 id='examples'>Examples</h2> <div class='num_example'> <h6 id='example'>Example</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/connecting+homomorphism'>connecting homomorphism</a> of the <a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+homology'>long exact sequence in homology</a> induced from short exact sequences of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo stretchy='false'>/</mo><msub><mi>A</mi> <mrow><mi>n</mi><mi>tor</mi></mrow></msub><mover><mo>→</mo><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⋅</mo><mi>n</mi></mrow></mover><mi>A</mi><mo>→</mo><mi>A</mi><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mi>n</mi><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> A/A_{n tor} \stackrel{(-)\cdot n}{\to} A \to A/(n A) </annotation></semantics></math></div> <p>is called a <em><a class='existingWikiWord' href='/nlab/show/diff/Bockstein+homomorphism'>Bockstein homomorphism</a></em>.</p> </div> <h2 id='properties'>Properties</h2> <h3 id='relation_to_homotopy_fiber_sequences'>Relation to homotopy fiber sequences</h3> <p>The connecting homomorphism in a <a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+homology'>long exact sequence in homology</a> induced from a short exact sequence <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mo>•</mo></msub><mover><mo>→</mo><mi>f</mi></mover><msub><mi>B</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>A_\bullet \stackrel{f}{\to} B_\bullet \to C_\bullet</annotation></semantics></math> is equivalently the image under the <a class='existingWikiWord' href='/nlab/show/diff/homology'>homology group</a> functor of the <a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>homotopy cofiber sequence</a> induced by <math class='maruku-mathml' display='inline' id='mathml_9bc56e281dfab5a9a27f1a363abe565041d23733_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>. This is discussed in detail at <em><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></em> in the section <em><a href='mapping+cone#HomologyExactSequencesAndFiberSequences'>homology exact sequences</a></em>.</p><ins class='diffins'> </ins><ins class='diffins'><h2 id='related_concepts'>Related concepts</h2></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+homology'>long exact sequence in homology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+chain+homology'>long exact sequence in chain homology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+generalized+homology'>long exact sequence in generalized homology</a></p> </li> </ul> </li> </ul></ins> <h2 id='references'>References</h2> <p>For instance section 1.3 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Charles+Weibel'>Charles Weibel</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/An+Introduction+to+Homological+Algebra'>An Introduction to Homological Algebra</a></em></li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on January 17, 2021 at 07:06:59. 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